Math 1320 sec 004
DATE: XXXXXXX
Name:
| {z by writing my name I swear by the honor code
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Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I may take off points if
I cannot see how you arrived at your answer (even if your final answer is correct).
• No calculators, notes or books are allowed for this test.
• Please keep your written answers brief; be clear and to the point. You may lose points for incorrect or irrelevant statements.
• Please turn off cellphones, take out headphones and sit every other seat.
• You have 50 minutes to work on this exam.
• Good luck!
Problem (value) Score
1 (20 points)
2 (10 points)
3 (12 points)
4 (20 points)
5 (18 points)
Total (80 points)

1.
( 20 points ) Short Answer. For True or False you must write ‘True’ or ‘False’ and justify your answer to receive credit.
a. ( 4 pts ) Consider a 2-D flat plate with constant density ρ = 1 bounded by x = 0, x = 1, y = 0 and y = f ( x ).
f ( x ) > 0 for all x . Write down an expression for the mass of the plate.
dx dt b. ( 4 pts ) True or False. The following differential equation is separable:
= e sin( t )+cos( x )
.
c. ( 4 pts ) pattern 1
− 2
, 2
− 3 / 2
Find an explicit formula for the n th term of the sequence that follows the
, 3
− 4 / 3
, 4
− 5 / 4
, ...
d. ( 4 pts ) If f ( x ) =
∞
P n =1 a n x n
, write down a series that represents R x
0 f ( t ) dt .
e. ( 4 pts ) How many terms are needed to approximate the convergent series
∞
P n =1
( − 1) n n using the n th partial sum to within 0 .
01?

2.
( 10 points )
∞
P k =3
1 k a. ( 5 pts )
−
1 k − 2
Write down the n th term in the sequence of partial sums for the series
. Does this series converge, and if so, to what?
b. ( 5 pts ) What is the interval of convergence for the power series
∞
P n =0 x n e n n +1
?

3.
( 12 points ) Determine if the following series converge or diverge and give a reason for your answer.
a. ( 4 pts )
∞
P n =1
7
1 n − 1 b. ( 4 pts )
∞
P n =1 cos(( n + 1) π )
3
1 n c. ( 4 pts )
∞
P n =1
√ n n − 1

4.
( 20 points ) a. ( 10 pts ) Solve the differential equation dy dx
= x y with y (0) = − 3.
b. ( 10 pts ) Tritium-3 decays exponentially according to m ( t ) = m
0 e
− kt with m measured in kilograms and t in years. If a sample has decayed to 95% of its original mass ( m
0
) after 1 year, what is its half-life?

5.
( 18 points ) a. ( 5 pts ) Set up, but do not solve, an integral that represents the volume of the region bounded by x = 1 + y
2
, x = 0, y = 1, and y = 2 rotated about the x -axis.
b. ( 5 pts ) Find the average value of f ( x ) = sin 4 x over the interval − π < x < π .
c. ( 8 pts ) A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is required to stretch the spring from its natural length to 6 in. beyond its natural length.