Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam 2
Spring 2000
Test Form A
1-10
/50
11
/10
12
/10
Part II is work out. Show all your work. Partial credit will be given.
13
/10
You may not use a calculator.
14
/10
15
/10
Part I is multiple choice. There is no partial credit.
TOTAL
1
Part I: Multiple Choice (5 points each)
1. Compute
;
There is no partial credit. You may not use a calculator.
1
"1
1 dx.
x6
a. 0
b. 2
5
c. " 2
7
d. " 2
5
e. Divergent
2. Find the solution to
a.
b.
c.
d.
e.
dy
dx
" xy satisfying the initial condition yŸ3 4. Then yŸ0 0
1
3
5
6
3. A tank contains 500 liters of water with 10 kg of sugar dissolved. Sugar water that contains
1 kg of sugar per liter of water flows into the tank at the rate of 7 liters per minute. Sugar
10
water that contains 1 kg of sugar per liter of water flows into the tank at the rate of 8 liters per
20
minute. The solution is kept thoroughly mixed and drains from the tank at the rate of 15 liters
per minute. Write down the initial value problem for SŸt , the amount of sugar in the tank at
time t.
a. dS
b.
c.
d.
e.
dt
dS
dt
dS
dt
dS
dt
dS
dt
10t 3 "
10
11 " 3S
10
100
3 " S
500
20
3 " 3S
100
20
11 t " S
10
500
S
with SŸ0 500
with SŸ0 10
with SŸ0 100
10
with SŸ0 1
50
with SŸ0 100
2
4.
Which integral gives the surface area of
the “spool” obtained by revolving the curve
y2 " x 1
0
for
"1 t y t 1
about the y-axis.
a.
b.
c.
d.
e.
;
;
;
;
;
1
"1
1
"1
1
"1
1
"1
1
"1
2=Ÿ1 y 2 1 4y 2 dy
2=y 1 4y 2 dy
=y 1 2y dy
2=Ÿ1 4y 2 1 y 2 dy
2=Ÿ1 y 2 1 2y 2 dy
5. Find the x-coordinate of the centroid (center of mass, x ) of the region between y
the x-axis for 0 t x t 2.
a. 4
5
4
b.
c. 8
5
d. 32
5
e. 2
3
x 3 and
3
6. Find the limit of the sequence a n
a.
b.
c.
d.
e.
lnŸn 2 1 .
lnŸn 0
ln 2
2
e
.
7. Which of the following series are convergent?
!
.
(i)
n1
a.
b.
c.
d.
100 n
n!
n1
2n
n 3n
both (i) and (ii)
(i) only
(ii) only
neither
8. Which of the following series are convergent?
!
.
(i)
n1
a.
b.
c.
d.
!
.
(ii)
n
n1
!
.
(ii)
n2
n
lnn
both (i) and (ii)
(i) only
(ii) only
neither
4
!
.
9. Compute
a. "20
n0
5 n1 .
4n
b. 20
9
c. 20
d. 25
e. divergent
!
.
10. The series
n1
a.
b.
c.
d.
e.
Ÿ" 1 n
n
is absolutely convergent.
is convergent but not absolutely convergent.
is divergent to ..
is divergent to "..
is divergent but not to o..
5
Part II: Work Out (10 points each)
Show all your work. Partial credit will be given.
You may not use a calculator.
11. Find the solution of
dy
" 3y
dx
!
.
12. Consider the series S
n1
6e "x satisfying yŸ0 5. Solve for y.
n " n1
n1
n2
!
3
a. Write out the third partial sum S 3
n1
n " n1
n1
n2
!
.
b. Compute the sum of the infinite series S
n1
and add it up.
n " n1 .
n1
n2
6
13. Find the arc length of the parametric curve x
t
0 and t
t3,
y
3t 2 between
12 .
!
!
.
14. Consider the series S
n2
.
a. Prove the series S
n2
1
.
nŸln n 2
1
nŸln n 2
converges. Be sure to name the convergence test
you use.
!
9
1
, find an upper bound
2
lnn
n
Ÿ
n2
|S " S 9 | in the approximation and justify your estimate.
b. If you approximate the series S by the partial sum S 9
for the error |R 9 |
7
15.
A triangular plate is suspended vertically
in water as shown. The height of the
triangle is 5 ft. Its base is 10 ft, and it is
submerged 3 ft into the water. Find the
force on one face of the plate.
(>g 62.5 lb3 )
ft
8