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04-05 Fall Final Exam

04-05 Fall Final Exam - Mathematics 104 Fall Term 2004-2005...

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Unformatted text preview: Mathematics 104 Fall Term 2004-2005 Final Examination January 14, 2005 Name Instructor Class hour There are eleven questions, weighted equally. Please answer them on these pages; no examination booklets should be used. This is a three hour examination. Calculators are not permitted. Write the pledge in full and sign it: I pledge mylhonor that I have not violated the Honor Code during this examination. The corrected examinatious can be obtained from your instructor. Do not call the department for grades. Your grade for the course is ' 2. For each of the following integrals, state whether it converges or diverges, and give your reasons carefully and clearly. 3. For each of the following series, state whether it converges or diverges, and give your reasons carefully and clearly. a. itarf (-31-) n=1 no b' fig} 1) n+lnn —-——-——-—-.»-———. . -——.-uu..... (e3 +e” — 2)\/5+a:§ +cos 1—cos5a: ' 4. Find lim 2-“) 5 . ._._.._.._..___ n . .. _ 5. Let flw) = e3 — 38in :5. Find f‘75)(0). (Use the formula relating the nth coefficient of a Taylor series to the nth derivative at the center.) ' d . f.- I 6. Find-an approximate'value for s/fi using a Taylor polynomial of order 2 and estimate the error. Show your work. it 7. For what values of :1: does 2 _(_5_:zi_—_l)_ converge? Give your reasons. -3. A curve is given in polar coordinates by r = 2— cos 9. Find the equation of the tangent linetothiscurveat9=§. 9. Write in Cartesian form, simplified, all solutions to (z -— l)3 + 8 = 0. 10. The region above the x-axis, below y~ - 2:3, for 2 < a: < 4 is revolved around the line a: = -—3. Find the volume. \ dy 11. Consider the initial value problem— do: -2:1:y = 21:, y(0) = 1. a. Solve it by finding an integrating factor. _......__._...—.__... _..._ .....——...—......—-—.—._._ Wan—m... . . _...__.. . no... .. ._ .___. b. Solve it by separating variables, and verify that your answer agrees with your answer to part a. ...
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