Name
ID
MATH 172
EXAM 2
Spring 1999
Section 504
1-10
/60
11
/10
12
/10
13
/10
14
/10
P. Yasskin
Multiple Choice: (6 points each)
7
; Ÿx 2 2x dx
1. Approximate
by using the trapezoid rule with 3 intervals.
1
a.
b.
c.
d.
e.
83
116
162
166
232
2
2. Given that the partial fraction expansion for x4 " 12 is
2
compute
; x4 " 12 dx .
x x
a. 2 tan "1 x 1
x C
b. tan "1 x 1
x C
2
c. tan "1 x " 1
x C
2
d.
"4x
2
Ÿx 1 2
2
x3
x
x
x2 " 1
x4 x2
2
x2
1
" 12 ,
x
C
e. None of These
1
;
3. Compute:
a. sin "1 Ÿ2 "
b.
=
2
1
x "1
2
1
dx
2
= " sin "1 Ÿ2 2
c. ln 2 3
d. ln 2 " 3
e. ln sec 2 tan 2
sec 1 tan 1
;
4. The improper integral
a.
b.
c.
d.
e.
2
1
1 Ÿx
" 1 2
dx
diverges to " .
converges to a negative number
converges to 0
converges to a positive number
diverges to .
;
5. The improper integral
a. diverges to
".
b. converges and is
c. converges and is
d. converges and is
e. diverges to
.
.
2
x2
1
dx
1 sin x
1
2
1
2
1
2
2
6. Which of the following is the direction field of the differential equation
a.
d.
b.
e.
dy
dx
xy 2 ?
c.
3
7. Solve the initial value problem
Then find
a. 0
b. 1
5
c. 4
5
d. 7
10
e. 1
yŸ2 .
8. The mass density of a 3 ft bar is
dy
dx
>
xy 2
1 x 2 lb
with the initial condition
ft
yŸ1 2 .
5
for 0 t x t 3. Find the center of
mass of the bar.
a. x 12
b. x 16
33
c. x 33
16
d. x 4
99
e. x 99
4
4
9. Find the arc length of the parametric curve
between t
a. 61
27
b. 125
9
125
c.
27
d. 122
3
250
e.
3
0 and t
x
2t 2
and
y
t3
3
1.
for
is rotated about the x-axis. The area of the
y x3
0txt2
resulting surface may be computed from the integral
10. The curve
a. ;
2
0
=x 1 9x 4 dx
2
b. ; 2=x 3 1 9x 4 dx
0
2
c. ; 2=x 1 x 6 dx
0
d. ;
2
0
=x 3 1 x 6 dx
2
e. ; 2=x 1 9x 4 dx
0
5
11. (10 points)
Find the partial fraction expansion for
(Do not integrate. HINT: Try x
12. (10 points)
Compute:
;
0, 1, "1.)
1
Ÿ1
" x 2 3/2
2x 2 " x 2 .
x3 x
dx.
6
13. (10 points)
condition
Solve the initial value problem
yŸ1 2.
dy
dx
2xy
1 x2
3x 2
1 x2
with the initial
7
14. (10 points)
A nuclear power plant went on line at the beginning of the year 1980. It
has produced isotope X at the rate of 10 kg
yr and the half-life of X is 20 yr. (So its
ln
2
decay constant is k . ) The plant stores all of the isotope X it produces. If
20
there was no isotope X at the beginning of 1980, how much isotope X will there be at
the beginning of the year 2000?
(6 points for the equations.)
8