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Unformatted text preview: Tufts University
Math 12 Department of Mathematics December 15, 2004
Final Exam 8:3010:30 a.m. Instructions. No calculators, notes or books allowed. You must show your work in the blue book
and cross out any work you do not want graded. Remember to Sign your blue book. With your
signature you are pledging that you have neither given nor received assistance on this exam.
Final course grades will be available to you 24 hours after they have been posted on SIS. Part I. On the inside of the from of your bluebook ( blue page), put answers only to the following
problems. No partial credit will be given. 1. (12 points) Fill in the blanks: (a) Choose one and complete: The sequence an 2 n 1+(_1)n {(1') diverges because (it) converges to 00 (b) Iff<rc> =2 2” n20n+3 (c) x/2 + x/2i in polar form is
(d) (\/2 + \/22')5 in polar form is (e) If \/2 + x/2i is one of the distinct fourth roots of a complex number 2, graph all four of the fourth roots of z on the first (blue) page of your bluebook.
2 5+3i :13”, then the derivative f’(0) is in the form a + bi is (f) Part II. For the problems in this part you must show your work in the blue book, giving full
justiﬁcation. Cross out any work you do not want graded. 2. (12 points) Determine whether each series is absolutely convergent, conditionally convergent,
or divergent. Justify your answer by citing and doing the appropriate convergence tests. (a) i (33” (b) :ln (2 + %) 72:1 3. (14 points) Find the radius of convergence and interval of convergence for the following power SCI'IeSéO (a: — 3)2n 00 (a: + 1)”
(a) ;W (b) ET. 4. (10 points) Use a series solution, y = 220:0 cnx” to solve the differential equation 3/ — 2y 2 0
by performing the following steps. (a) Express cn+1 in terms of C”.
(b) Write cn in terms of co.
(c) Write out the series solution to the differential equation.
((1) Identify the function to which the series in (4c) converges when y(0) = 3.
5. (7 points) Find the solution to the following differential equation for 0 < at < 7r / 2
y'sinm=(1+y)cosx, y(7r/3)=1. Do not use series methods. Write your solution as a formula for y in terms of at. Please turn over. 6. (18 points) Evaluate the following integrals. . 3:0 1
(a) /.7:51n2.77dac (b) /md$ (C) $2 f1+$2 dac 7. (15 points) A curve, C, is given by the parametric equations:
02 {at = 1 +sin2t, y = sin4t}. (a) Eliminate the variable t to ﬁnd a Cartesian equation. Graph the curve C using the
Cartesian equation to help you. (b) On the graph in (a), indicate the path a particle would traverse on C as t increases from
77/2 to 77. Label the starting point and ending point. Indicate with arrows, how the path
is traversed as t increases from 77/2 to 77. (c) Set up, but DO NOT EVALUATE, an integral for the arc length of C for 77/2 3 t g 77. 8. (6 points) Graph the curve 7“ = 77 — 0, 0 g 0 g 277. Label where the curve starts, ends, and
crosses the horizontal and the vertical axes. Use arrows to indicate the direction along the
curve for increasing 6. 9. (6 points) Set—up, but DO NOT EVALUATE, an integral(s) for the area in the ﬁrst quadrant
enclosed by the curves 7“ = sin0 and 7“ = sin 20, which is the shaded region below. ...
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