Name
ID
MATH 172
EXAM 2
Section 502
a.
b.
c.
d.
e.
/50
11
/15
12
/10
13
/10
14
/15
P. Yasskin
Multiple Choice: (5 points each)
1. Compute
Fall 1998
1-10
4
; lnx dx
1
x
4 ln 4
4 ln 4 " 2
4 ln 4 " 4
4 ln 4 " 12
2 ln 2 " 8
2. Approximate the integral
6
; x 3 dx
by using the Midpoint Rule with 3 rectangles
0
with equal widths
a. 162
b. 306
c. 324
d. 360
e. 648
1
3. Compute
;
=/2
0
sin 4 2 cos 2 d2
a. " 1
3
b. " 1
c. 1
6
d. 1
5
e. 1
3
5
4. Compute
;
=/4
0
tan 3 2 sec 2 2 d2
a. " 1
b. 1
4
4
c. " 1
d. 1
3
3
e. " 1
2
5. Compute
;
.
e
a. ".
1
dx
xŸlnx 2
b. " 1
e
c. 1
e
d. 1
e. .
6. Compute
a. ".
b. "1
;
e
1
1
dx
xŸlnx 2
c. 1
e
d. 1
e. .
2
7. Which of the following is the direction field of the differential equation
a.
d.
b.
e.
dy
x 2 y?
dx
c.
3
1 " x2
x Ÿ1 x 2 8. Given that the partial fraction expansion of
1 " x2
12 " 2 2 ,
1x
x Ÿ1 x 2 x
2
compute:
2
;
is
1 " x2
dx
x Ÿ1 x 2 2
a. " 1 C
2x
4x
C
b. " 23 2 2
x
Ÿ1 x c. ln x 2 " lnŸ1 x 2 C
d. ln x 2 " 2 lnŸ1 x 2 C
e. " 1
x " 2 arctan x C
9. Solve the differential equation
Then find
a. "1
b. 0
c. 1
d. " 2
3
e. 3
2
with the initial condition
yŸ0 2.
y
dy
" x 2x 2
dx
with the initial condition
yŸ1 3.
yŸ1 .
10. Solve the differential equation
Then find
a. 2
b. 4
c. 6
d. 8
e. 12
dy
xy 2 y 2
dx
yŸ2 .
4
11. (15 points) Compute
1
; Ÿt 2 " t e 2t dt
0
5
12. (10 points) Compute
;
2
1
x2 " 1
x
dx
6
13. (10 points) Find the partial fraction expansion for
(Do not integrate.)
Hints:
Try:
x 1,
x0
and
Ÿx
4x " 6
.
" 1 2 Ÿx 2 1 d .
dx
7
14. (15 points) A tank initially contains
of sugar water with
of dissolved
12L
8gm
sugar. Sugar water that contains
of sugar per liter is poured into the tank at
2gm
the rate of
The solution is kept thoroughly mixed and drains from the tank
3L/min.
at the rate of
How much sugar is in the tank
3L/min.
(a) after
(b) after
and
(c) asymptotically after a large time?
t min
16min
8