MAT2500-01/04 19S Final Exam Print Name (Last, First) _________________________________|__
Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use EQUAL signs and arrows when appropriate. Always SIMPLIFY expressions. LABEL parts of problem. BOX final short answers. Keep answers EXACT (but give decimal approximations for interpretation if appropriate). Indicate where technology is used and what type (Maple, GC). Use technology to evaluate any integrals you set up. JUSTIFY any numbers or equations that play a role in your calculations.
Recall: 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. Green's Theorem: for a counterclockwise oriented curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn which is the boundary of R:
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
a) Use Green\342\200\231s Theorem to evaluate 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 , where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ== is the circle 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 with counterclockwise orientation. Annotate your diagram of the curve and its interior to describe your double integration iteration. Integrate by hand.b) Repeat for the circle 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 with the counterclockwise orientation.
c) Using standard polar coordinates to parametrize the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==, evaluate the line integral around it directly, evaluating the final integral in Maple.2. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkmbW92ZXJHRiQ2JS1JI21pR0YkNilRIkZGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkjbW9HRiQ2L1EtJlJpZ2h0QXJyb3c7RidGNy8lLHBsYWNlaG9sZGVyR0Y0L0Y+USdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zmbi9GWUY0LUkobWZlbmNlZEdGJDYkLUYjNiktRi82JVEieEYnRjUvRj5RJ2l0YWxpY0YnLUZENi1RIixGJ0ZJL0ZMRjkvRk9GNC9GUUY5L0ZTRjkvRlVGOS9GV0Y5L0ZZRjlGWi9GaG5RLDAuMzMzMzMzM2VtRictRi82JVEieUYnRjVGYm9GZG8tRi82JVEiekYnRjVGYm9GOkZJRkktRkQ2LVEiPUYnRklGZ28vRk9GOUZpb0Zqb0ZbcEZccEZdcC9GZW5RLDAuMjc3Nzc3OGVtRicvRmhuRltxLUZbbzYmLUYjNjAtSSVtc3VwR0YkNiVGX28tRiM2JS1JI21uR0YkNiRRIjJGJ0ZJRjpGSS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRkQ2LVEifkYnRklGZ29GaXBGaW9Gam9GW3BGXHBGXXBGWkZnbkZgcEZkb0ZgcEZdckZjcEZkb0ZdckZjcEZdckZhcUY6RklGSS8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGXXJGZG9GOkZJLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiOkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GOTYuUSJ+RidGMkY8Rj5GQEZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBGVS1GLDYmUSJ4RidGL0YyRjUtRjk2LlEiPUYnRjJGPEY+RkBGQkZERkZGSEZKRkxGTy1GLDYmUSRjb3NGJy9GMEY0RjJGPC1JKG1mZW5jZWRHRiQ2JS1GIzYlLUYsNiZRInRGJ0YvRjJGNUYyRjxGMkY8LUY5Ni5RIixGJ0YyRjxGPi9GQUYxRkJGREZGRkhGSkZUL0ZQUSwwLjMzMzMzMzNlbUYnLUYsNiZRInlGJ0YvRjJGNUZaLUYsNiZRJHNpbkYnRmpuRjJGPEZbb0Zjby1GLDYmUSJ6RidGL0YyRjVGWkZccEZbb0Zjb0Zgb0ZaLUkjbW5HRiQ2JVEiMEYnRjJGPC1GOTYuUSMuLkYnRjJGPEY+RkBGQkZERkZGSEZKL0ZNUSwwLjIyMjIyMjJlbUYnRlYtRmNwNiVRIjJGJ0YyRjxGUS1GLDYmUSUmcGk7RidGam5GMkY8RjJGPA==
Set up the line integral of this vector field along this parametrized curve, and then use Maple to evaluate it exactly after simplifying the integrand.3. 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) Evaluate the divergence of LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21pR0YkNihRIkZGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjEvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUS0mUmlnaHRBcnJvdztGJ0Y0LyUscGxhY2Vob2xkZXJHRjEvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYvRlNGMQ==.b) Show that curlLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNiYtSSZtb3ZlckdGJDYlLUkjbWlHRiQ2KFEiRkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGOS8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JI21vR0YkNi9RLSZSaWdodEFycm93O0YnRjwvJSxwbGFjZWhvbGRlckdGOS9GQFEnbm9ybWFsRicvJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Znbi9GWkY5LUY0NiNRIUYnRjxGS0Y8RkstRkY2LlEiPUYnRjxGS0ZNRk8vRlJGPkZTRlVGV0ZZRmVuRmhuLUYxNiUtSSNtbkdGJDYnUSIwRidGN0Y8L0ZARkRGQi1GRjYvRkhGPEZJRksvRk5RJnVuc2V0RicvRlBGXHBGUS9GVEZccC9GVkZccC9GWEZccC9GWkZccC9GZm5RJjAuMGVtRicvRmluRmNwRmpuRjxGSw==, i.e., it is a conservative vector field.c) Find a potential function 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 for LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21pR0YkNihRIkZGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjEvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUS0mUmlnaHRBcnJvdztGJ0Y0LyUscGxhY2Vob2xkZXJHRjEvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYvRlNGMQ==.d) Use the potential to evaluate the line integral 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 from the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNiktSSNtbkdGJDYlUSIxRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiMkYnRjRGN0Y6LUYxNiVRIjNGJ0Y0RjdGNEY3RjRGN0Y0Rjc= to the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNiktSSNtbkdGJDYlUSIyRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiM0YnRjRGN0Y6LUYxNiVRIjRGJ0Y0RjdGNEY3RjRGN0Y0Rjc=.e) Evaluate 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 along the straight line from between the same initial and final points. Use Maple to evaluate the integral.
Do your results for d) and e) agree as they should? Do you know what year the Wildcats won their first NCAA championship?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The plots are not meant to be done with this sophistication by students who have only elementary Maple skills.1. Green's Theorem.a) Use Green\342\200\231s Theorem to evaluate 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 , where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ== is the circle 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 with counterclockwise orientation. Annotate your diagram of the curve and its interior to describe your double integration iteration.b) Repeat for the circle 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 with the counterclockwise orientation.
c) Using standard polar coordinates to parametrize the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==, evaluate the line integral around it directly, doing the final iterated integral in Maple.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRmVuL0ZORmZuLUZQNiYtRiM2Ky1JJW1zdXBHRiQ2JUZULUYjNiYtSSNtbkdGJDYkUSIyRidGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRmVuRmFvRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGaW5GVy1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GaXBGVEZfcC1GZ282JUZpbi1GIzYlRltwRi9GMkZicEZcb0Y5RjkvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRl9wRlxvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkmbWZyYWNHRiQ2KC1JI21vR0YkNi1RKyZQYXJ0aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRi1GIzYmLUYvNi1GMUYyL0Y2USZmYWxzZUYnL0Y5Rk4vRjtGTi9GPUZOL0Y/Rk4vRkFGTi9GQ0ZORkRGRy1GLzYtUSJ+RidGMkZNRk9GUEZRRlJGU0ZURkRGRy1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGTkZVLUZZNiVRIkZGJ0ZmbkZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYnRlgtRi82LVEiLEYnRjJGTS9GOUZobkZQRlFGUkZTRlRGRC9GSFEsMC4zMzMzMzMzZW1GJy1GWTYlUSJ5RidGZm5GaW4vJStleGVjdXRhYmxlR0ZORjJGMi1GaW82Ji1GIzYlLUkjbW5HRiQ2JFEiMkYnRjJGZnBGMkYyLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUYvNi1RKiZ1bWludXMwO0YnRjJGTUZPRlBGUUZSRlNGVC9GRVEsMC4yMjIyMjIyZW1GJy9GSEZqcS1GLDYoRi4tRiM2JkZLRlVGY3BGMkZbb0Zeb0Zhb0Zjb0ZVRmVvRmhvLUZpbzYmLUYjNiUtRl1xNiRGXW9GMkZmcEYyRjJGYHFGY3EtRi82LVEiPUYnRjJGTUZPRlBGUUZSRlNGVC9GRVEsMC4yNzc3Nzc4ZW1GJy9GSEZqckZmcS1JJW1zdXBHRiQ2JS1GWTYlUSJyRidGZm5GaW4tRiM2JUZccUZmbkZpbi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGZnBGMg==a)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b)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNiQtSSNtbkdGJDYkUSIwRidGMkYyLUYjNiQtSSNtaUdGJDYlUSUmcGk7RicvJSdpdGFsaWNHUSZmYWxzZUYnRjJGMi8lMXN1cGVyc2NyaXB0c2hpZnRHRk8vJS9zdWJzY3JpcHRzaGlmdEdGTy1GLDYnRi5GSi1GIzYnLUZNNiRRIjRGJ0YyLUYvNi1RIn5GJ0YyL0Y2RlgvRjlGWC9GO0ZYL0Y+RlgvRkBGWC9GQkZYL0ZERlhGRUZILUZTNiVRJGNvc0YnRlZGMi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZTNiVRInRGJy9GV0Y8L0YzUSdpdGFsaWNGJ0YyRjJGMkZZRmVuLUYvNi1RKCZtaW51cztGJ0YyRmFvRmJvRmNvRmRvRmVvRmZvRmdvL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRmpwLUklbXN1cEdGJDYlLUZTNiVRInJGJ0ZjcEZkcC1GIzYlLUZNNiRRIjNGJ0YyRmNwRmRwRllGXm8tRi82LVEwJkRpZmZlcmVudGlhbEQ7RidGMkY1RjgvRjtGN0Y9L0ZARjdGQUZDRkVGSEZfcUZeb0ZncUZgcC8lK2V4ZWN1dGFibGVHRlhGMg==c)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=2.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=MFNWtKUb<ob<R=MDLCdNVZ:L:PV>D:;b\\Qs<?\\ZRLCTJcDXoHuuV<;DOehyIxuSsWGV@gYOmB^=uN;T:=DK=R?iYcsdBcHNatLyYceV_Uye_YwwTiiIewViwVegifEhLUSLEsWET;ER?]DKERJ;TJkB^]TPEu`uUxyvHaCj;CBkdZ=wqydWkImmvVYXxuVuwxqusp[gL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conservative vector 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you have completed the exam, please read and sign the dr bob integrity pledge if it applies and hand in with your answer sheets as a cover page, with the Lastname, FirstName side face up: "During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: