Name
ID
MATH 172
EXAM 3
Section 502
Fall 1999
1-10
/80
11
/15
12
/15
P. Yasskin
Multiple Choice: (8 points each)
1. Which of the following is the direction field of the differential equation
a.
b.
d.
e.
2. The limit lim
nv.
dy
xy ?
dx
c.
4Ÿ2 n
n
Ÿ5 a. converges to 0.
b. converges to
4
.
1" 2
5
c. converges to 4.
8
5 .
d. converges to
1" 2
5
e. diverges.
3. lim
1 " 12
nv.
n
a.
b.
c.
d.
e.
3n 2
e "6
e6
1
e
e "3
1
!
.
4. The series
Ÿ" 1 n1 n
2
3n
n1
a. diverges by the n th -Term Test.
b. converges to 2
c. converges to 2
5
d. converges to " 3
5
e. diverges by the Alternating Series Test
!
.
5.
n1
a.
b.
c.
d.
e.
n1 " n2
n
n1
0
1
2
3
.
!
.
6. The series
n1
a.
b.
c.
d.
n!
2n
converges by the Integral Test.
diverges by the Integral Test.
converges by the Ratio Test.
diverges by the Ratio Test.
e. converges by Comparison with
!
.
n1
!
.
7. The series
n2
1 .
2n
2
n3 n
a. diverges by the n th -Term Test.
b. converges by the Ratio Test.
c. diverges by the Ratio Test.
!
!
.
d. converges by Comparison to
.
e. diverges by Comparison to
n2
n2
2
n3
2
n
2
!
.
8. The series
n2
a.
b.
c.
d.
e.
9. The series
!
n1
Ÿ" 1 n
n1
is
Absolutely Convergent
Conditionally Convergent
Convergent for Even n, Divergent for Odd n
Convergent for Odd n, Divergent for Even n
Divergent
!
.
10. The series
n1
a.
b.
c.
d.
e.
n1
converges by the Integral Test.
diverges by the Integral Test.
converges by the Ratio Test.
diverges by the Ratio Test.
diverges by the n th -Term Test.
.
a.
b.
c.
d.
e.
1
Ÿ" 1 n 3/2
n
is
Absolutely Convergent
Conditionally Convergent
Convergent for Even n, Divergent for Odd n
Convergent for Odd n, Divergent for Even n
Divergent
3
11. (15 points) Solve the differential equation 1
x
3
dy
3xy e "x with yŸ0 2
dx
4
12. (15 points) A tank contains 5000 gal of water. Initially, there are 10 lb of salt dissolved
in the water. Salt water containing 0.03 lb of salt per gal is added to the tank at the rate
of 2 gal per hour. The solution is kept mixed and is drained at the rate of 2 gal per
hour. Let SŸt be the amount of salt in the tank at time t.
a. State the differential equation and the initial condition satisfied by SŸt .
b. Solve this initial value problem.
c. At large times, what is the asymptotic amount of salt in the tank.
5