Solutions Final Exam

Solutions Final Exam - ME 391 Final Exam April 28, 2008...

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Unformatted text preview: ME 391 Final Exam April 28, 2008 Spring 2008 Instructions: 1. You may use an 8.5x11” sheet of formulas, theorems, hints (No example problems) 2. No Calculators are allowed 3. Show all your work clearly and with reasoning. 4. 200 pts total + 10 extra credit pts Laplace Transforms Problem 1a: Solve using Derivatives of Transformszx {t2 cos t} (10 pts) A ‘ A A: K {k LCQE‘ ( (dgu) (Cs) (xi/3905329: s \, .4. (Ls: ) 244-53 37'“ ’ -: Ag— ( as (31ml 1b: Solve using Convolutions: X 4 43- (10 pts) 8 -| X"{:~?¥z*yl= or?“ {can as)? r m b foxe’l‘e" (KT) ‘ be_x Yoxez’t‘df 74" 'x e "l X P 7-06 [ 1: Se‘Sex Z 9—,... Problem 3: Use Laplace Transform to solve the NP Linear Differential Equation: (20 pts) y”+5y’+4y=0 y(0)=1.y’(0)=0 Vt); ‘—‘ a“ :(53- skd‘ificfi ' garb)— szaeflfi_ 6“ Co) _ 2; (OX ’awfl -l— {33:0 €o\uz Cw ’6’ 7 A J— , .L : 5::‘3’$+4 3 6H 3 (3+4 ’6 4- -4t ‘ ,‘i — _ (Ase Mia ‘6f 3‘3 35 Problem 2) Find the Laplace Transform of the given periodic function: (20 pts) 10“) T} 2a, Matrices: Problem 4: Solve the following system using Gauss—Jordan elimination: (20 pts ea) x1 + 2X2-X3 = O 2X1+X2+2X3=9 X1-X2+X3=3 ’l O 'ZQ‘+QZ-§QZ \ ‘2‘ "\ 4 z 5/5 a '4’3 '5 Problem 5a: Evaluate the determinant of the following Matrix by reducing to Echelon Form (REF). (15pts) I z 3 A" 4 5 o ’l 8 ‘l ~4£,+{ZZ->Q7‘ ' 2— 3' ,l 1 Z 3 - 3 Q1621. -12\+Q§§ O 3 ‘(J (a O \ 2.. O ab ,‘2 ~2. (17345—9 3 o o o BET : fl 5b: What is the rank of this Matrix? (5 pts) (EAJL : Z (209) (5‘3) Problem 6a: Using the Identity method, find the Inverse of the following Matrix: (10 pts) '2. ‘ l A‘ *6 ’5 o I \ \ Z . 1 t 00 t o o t ?\~13—5?' ’S ‘3 0 O ‘0 ~5 ‘1 O O \ l t O O\ l t ‘ D \ O O l 0‘! I i O O ‘ o*$05\-5 ’3fl10t0-5‘/S t \ \ o o | ‘ \ t O I c) O (O " l O O l .. ——l 5‘ *‘ ,§ 0 s o 3-4, /3 g gigs-3Q O i 0 g 0 t l “t o 2_ o o I 2/?) -1 ‘ 01 -5: A 7 '5/3 '4) /3 1 ya /3 y: 63: What is its Transpose, AT? (10pts) -6 Z/ '- /3 3 T- o -! c A ‘ _ E /3 I '3 l/ Problem 7: Determine the Inverse of the following Matrix by first finding its Adjoint: ( a i (20pts) A: L“ \ L 2, a 4 3ch A = i (4-0—344—4} + 1(3-23 :: "Z. “' O +| :: "t “M i '2— “z i ‘L C ' M ‘ Cu: :42 an: _ 0 IS” :_ 1 ’5 4 7. 4 - a 3 i 74'" - 1“: Cuslu)g ‘\:’q 622:.l ‘ l :2 0234 1; 3 3H 3 i 3+2 . i - C :“1 I =5 C :‘l g- 3.7:! I Z 52' a, a ’ l 3+3 ,'\ i 3 ~Z 19 i Cas’ \[ ‘ = “2. Maciw .C= «a: 7. 3 Mm‘viic «5' -i’ «r ’ M AvC -— Z “i b 3 z ’i i '5 ’2. , .i‘ __L -2 '01 ‘3 Wiggins $4M =—i (0 7- 4) i 3 'Z 0i ‘6 -i Z A : 0 "Z 1 _\ “S 2- Problem 8: Determine the Eigenvalues of the following Matrix: (10 pts) ( a Afloaw A13“:- l'i)‘ 2' 3 Z-A AZ- 3A '4 (A+\3(/\'43 A‘ = “i ) A734 8b: Determine the Eigenvectors of the previous Matrix: (10pts) Afiw kn»? \ \ I—L'Q z 1 Z 1‘ ‘ z 3 3 a 2“! 3 3 l i 31‘“R3 0 0 Z - 3 Z. Rival -3 2- JL| : 'i—L z—‘i ’ 3 Q o O = \ Problem 9: Solve the following system using Cramer’s Rule: (20pts) x+2y =0 -x+y+ z=1 x+2y+ 32:0 2 lo l 7. o O A: ‘\ l l l l 7, 3 0 \ l "l l : Auk-H‘ZJJ. ‘3‘4 0 Z O _ l l 421%): 1‘: I l ':2 o S Z'b X; alt/Wk" 5‘ 0 Z 5 I O O l \ a ’L “ l l :‘ 1 Z 5 8:? -; 3 I o 2: 0 ~ l 2 o l 7. ’/'\ l l :‘IIZ :g Mayo 1 2. O (X/Wl13:< 3,5mfl Problem 10: Use the Taylor Series method to solve the following Initial Value Problem: (state the first 5 terms) (20 pts) y’—y2='x y(o)=1 I; V37— - X “(0) NCO 50m (Kg/park; (a; (03+ ‘(o)x+ %__3_(_)x3.+ W . (E . “(032% [=- 1’X ‘(oht 3., °\ 23 a“: 249(l—\ v————-———-'I \AHCQ, Z-l =( 3:; Zara: 2%) ————a 23%)): 2+ 7. =4 3 " 2% *“m”-->z)‘“‘(0‘=6 + ° “4 Extra Credit: Find the Fourier Sine Series for (10 pts) f(x) = 1 on (0, 5) 1 °’ Wm LP LIX]: 2 ’0 SM — X A:\ h P 7- " M¥ L“: F? Mask 7; m f 0 Z. 3 \ hfi‘x ‘3“ :1 g Sb 0‘ Qua g M , z 5 «MW 5 ,. 6 NW 005 5 O 2. 2. r; "‘ (‘ wgh’fi‘] : —— [l_(_‘\k] NW V11?" £00: Er. [\— (“(3 ]$\vx 6 4 1:] .L 5m .5. aw 3:6(6m 5+33m?+5'6u~? ...
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This note was uploaded on 07/25/2008 for the course ME 391 taught by Professor Blazer-adams during the Spring '08 term at Michigan State University.

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Solutions Final Exam - ME 391 Final Exam April 28, 2008...

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