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Unformatted text preview: Mathematics 104
Fall Term 20042005
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. Findan 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 xaxis, 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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 '08
 EdwardNelson
 Calculus

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