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WUSTL ESE 318 - Exam 1 Solutions n grading

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ESE 318-02 Exam 1 Feb. 11, 2016 NAME:__69 students took the exam___ All notation should conform to the Zill text. U(t-a) is the Unit Step function. 1. (5 points each) Find each indicated Laplace transform. L e−3tcosh 5t{ }= L cosh 5t{ }s=s+3=s + 3s + 3( )2− 25=s + 3s2+ 6s −16L t −τ( )2cosτdτ0t∫⎧⎨⎩⎫⎬⎭= L t2{ }L cosτ{ }=2s3ss2+1=2s2s2+1( )L tetU t −1( ){ }= e−sL t +1( )et+1{ }= e−s+1L tet+ et{ }= e−s+11s −1( )2+1s −1⎡⎣⎢⎢⎤⎦⎥⎥=e− s−1( )ss −1( )2also = −ddsL etU t −1( ){ }= −ddse−sL et+1{ }⎡⎣⎤⎦= −ddse−s+1s −1⎡⎣⎢⎤⎦⎥= −s −1( )−e−s+1( )− e−s+1s −1( )2=e− s−1( )ss −1( )2 Grading: 14 students got all 15 points. With so few points and such short problems, partial credit was close to all-or-nothing. But I did allow some partial credit, as follows: Part (a): Left “k” in, or wrong k, instead of 5: 4 pts. Sign wrong (or did cos, not cosh): 3 pts. Part (b): Left “k” in instead of 1: 4 pts. Used “t” instead of “s”: 3 pts. Had τ in the answer (big no-no): 1 pt. Took transform of 1 or t instead of t2: 1 pt. Part (c): Single sign error: 4 pts. Multiple sign errors: 3 pts. Had e-s instead of e-s+1: 2 pts. Sort-of “close”: 1 pt.ESE 318-02 Exam 1 Feb. 11, 2016 2. (5 points each) Find the inverse Laplace transform, f(t), of each Laplace transform. ( )( ) ( )( )[ ]( ) ( )( )[ ]( )( )( ) ( ) ( )( )( ) ( ) ( )( )( )( )ttfsssBACBAsCsBsAAssCBssAssssFtttfttftsssdsdFssFtetetfsssssssssFtettettfsesesFttttttsscos3413434014114sinsin1111111arctan4sin24cos16144816111618117293321321222121222222222222223223223333−=→+−=−====++=++++++=++==⇒−=−→+−=⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=⎟⎠⎞⎜⎝⎛=+=→++++++=++++=+++=−−=−=→−=−=−−−−=−−UU Grading: 7 students got all 20 points. Only 4 students got off the 1st page unscathed. Again, with so few points and such short problems, partial credit was close to all-or-nothing. But I did allow some partial credit, as follows: Part (a): Left off the factor of ½ or made it 2: 4 pts. Left of the U(t-3): 2 pts. Didn’t evaluate at t-3, or did that only partially: 2 pts. Part (b): Wrong coefficient on the sin term (not a 2): 3 pts. Wrong coefficient in sin & cos (not a 4): 2 pts. Multiple such errors, including wrong coefficient in exponent: 1 pt. Part (c): Wrong sign: 4 pts. Multiplied by t instead of dividing: 2 pts. Multiple such errors: 1 pt. Part (d): Sign: 4 pts. PFE arithmetic/algebra error: 3 pts. Got sin instead of cos: 2 pts.ESE 318-02 Exam 1 Feb. 11, 2016 3. (10 points) Write the subsidiary equation (Laplace domain in Y(s)) of the differential equation below, where f(t) is the periodic function graphed below it. Do not solve or simplify. ( ) ( )3010)(43 =ʹ==+ʹ+ʹʹyytfyyy ( )( )( )( ) ( )asasasasasatstasastaseseseYsYsYseseseedteesF−−−−−=−−−−+=−−=+−+−−−−=⎥⎦⎤⎢⎣⎡−−=−=∫111141331111112220202 Grading: I awarded 5 points each to the “left” side and the “right” side. Only 16 papers received full credit. Left side errors: • Minor “sloppy” error: -1 pt. • Sign error, didn’t distribute the 3 in y’ or left it off: -2 pts. each. • Did not substitute values for y(0) and y’(0): -2 pts. • Totally neglected y’, multiple errors: -4 pts. Right side errors: • Minor “sloppy” error: -1 pt. • Set up integral OK, but didn’t evaluate, or made an error: -2 pts. • Wrote integral with wrong limits: -3 pts. • Wrote integrand using “f(t)” or U(t) and didn’t evaluate: -4 pts. • Did not use formula for periodic function at all: -5 pts. A large number of students did not realize, or forgot, there is a formula for periodic functions, and it was on the Formula Sheet.ESE 318-02 Exam 1 Feb. 11, 2016 4. (10 points) Find y(t). ʹy + 2 y + yτ( )dτ0t∫= 0 y 0( )=1 sY −1+ 2Y +1sY = 0s + 2 +1s⎛⎝⎜⎞⎠⎟Y =1s2+ 2s +1( )Y = s ⇒ Y =ss2+ 2s +1=ss +1( )2Y =s +1−1s +1( )2=1s +1−1s +1( )2y t( )= e−t− te−t= e−t1−t( ) Grading: 32 students got all 10 points. • Correct subsidiary equation: at least 3 pts. o If integral not handled correctly: 0 pts. • Correct Y(s): at least 6 pts. o Sign error in y(t): 9 pts. o Extra factor of ½ in y(t): 8 pts. o One of the terms in y(t) correct: 7 pts. o No inverse or totally wrong inverse: 6 pts. • “Sloppy” error in algebra that carried through, but otherwise OK: 7 pts.ESE 318-02 Exam 1 Feb. 11, 2016 5. (15 points) Find y(t). Also, draw a rough sketch of y(t) for 0 < t < 2π. ( ) ( ) ( )10004 =ʹ=−=+ʹʹyytyyπδ s2Y −1+ 4Y = e−πss2+ 4( )Y =1+ e−πsY =1s2+ 4+ e−πs1s2+ 4=122s2+ 4+12e−πs2s2+ 4y t( )=12sin 2t +12sin 2 t −π( )U t −π( )=12sin 2t +12sin 2tU t −π( )=12sin 2tsin 2t⎧⎨⎪⎩⎪0 ≤ t <πt ≥π Grading: 10 points for finding y(t), 5 points for the sketch. 21 papers got full credit. (I did not require you to see that sin2(t-π) = sin2t.) • Finding y(t): o Off by a factor of 2, or a sign error: -2 pts. o More serious errors: -3 pts each. • Sketch. o Completely missing: -5 pts. o Didn’t match your y(t) at all: -3 to -4 pts. o Bad period, missing scale, other “slightly off” mistakes: -2 pts. • Severely messed up generally: o Multiple and/or serious errors throughout: 5 pts. o No real idea how to handle e−πs: 4 pts. o Had no idea how to get past line 3 above: 3 pts.ESE 318-02 Exam 1 Feb. 11, 2016 6. (10 points) Given the system of equations ( )( )103255235212121=−+=−+=+xaxxaxaxx (a) Find all values of a, if any, for which there are no solutions. (b) Find all values of a, if any, for which there is exactly one solution. (c) Find all values of a, if any, for which there is an infinite number of solutions. ( )( ) ( )21112000510515105105115330102205125105320155320511032552351==−=−≠⎩⎨⎧=⇒⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛++→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−−−=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−−−→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−=nBArankaifaifArankaaaaaaaaaaaaaaaaBA (a) 1−=a gives no solution. (b) Any1−≠a gives exactly one solution. (c) No a exists which yields an infinite number of solutions. Grading: 53 papers got this correct. As for the rest: • Got the value of 1 instead of -1, but the rest OK: 8 pts. • Got (a) & (b), or (b) & (c) conclusions


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