Tufts University
Math 12
Department of Mathematics
May 5, 2006
Final Exam
8:3010:30 am
Instructions:
Final course grades will be available to you 24 hours after they have been posted on SIS. No
calculators, notes or books allowed. Remember to sign your blue book. With your signature you are pledging
that you have neither given nor received assistance on this exam.
Part I.
On the first page of your bluebook, put answers only
to the following problem. No partial credit is
available.
1. (7 points total)
a) (3 points) Give the three cube roots of

8
i
in polar form.
b) (1 point) Plot the cube roots of

8
i
in the complex plane.
c) (3 points) Find the complex number
z
satisfying
z
+
iz
= 2

i
. Express your answer in the form
z
=
a
+
bi
.
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. (18 points) Evaluate
a)
t
2
ln
t dt
b)
1
x
2
(
x

1)(
x
+ 1)
dx
c)
sin
2
(3
θ
)
dθ
3. (10 points) Justify your answers to the following questions by citing and performing the appropriate
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 Spring '08
 GARANT
 Math, Complex number, appropriate convergence tests

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