Name
MATH 172
FINAL EXAM
P. Yasskin
Multiple Choice: (5 points each)
a Ÿ2, 0, 1 a.
b.
c.
d.
e.
a.
b.
c.
d.
e.
/10
16
/10
17
/104
and
b Ÿ"1, 2, 1 then
" 3b 2a
and
b Ÿ"1, 2, 1 then
a • b and
b Ÿ"1, 2, 1 are two edges of a triangle, find the area of
Ÿ1, "6, 5 Ÿ7, 6, "1 Ÿ7, "6, "1 Ÿ"1, 6, "5 Ÿ"2, "3, 4 Ÿ"2, 3, 4 Ÿ"2, 0, 1 Ÿ4, 0, "1 Ÿ4, 3, "2 a Ÿ2, 0, 1 3. If
15
Ÿ1, 6, 5 a Ÿ2, 0, 1 2. If
/70
Spring 1999
Section 504
1. If
1-14
ID
the triangle.
a.
b.
c.
d.
e.
3
3
2
1 29
2
29
2
29
1
u 4. If
v
is
a.
b.
c.
d.
e.
2 , "1, 1
and
2
v Ÿ0, 1, "1 then the angle between
u
and
30°
45°
60°
120°
135°
!
.
5. The series
n0
Ÿ" 1 Ÿx
2 n1
n
" 2 n
is the Taylor series about
x2
for
(Hint: Just sum the series.)
a. 1
x
b. 2
x
1
x"2
2
d.
x"2
1
e.
2Ÿx " 2 c.
!
.
6. Find the radius of convergence of the series
n0
a.
b.
c.
d.
e.
Ÿ" 1 Ÿx
2 n1
n
" 2 n.
0
1
2
4
8
2
!
.
7. The series
n1
n!
n1
a.
b.
c.
d.
e.
is
Divergent by the n th term Divergence Test
Convergent by the Ratio Test
Divergent by the Ratio Test
Convergent by the Integral Test
Divergent by the Integral Test
8. Compute
3
sinŸ2x " 2x 8x
3
lim
xv0
x5
a. ".
b. 0
c. 4
3
d. 2
5!
e. .
9. Compute
;
Ÿ2x
3
" 1 sinŸx 4 " 2x dx
a. 2 cosŸx 4 " 2x C
b. "2 cosŸx 4 " 2x C
c. 1 cosŸx 4 " 2x C
2
d. " 1 cosŸx 4 " 2x C
2
e. cosŸx 4 " 2x C
3
10. Compute
;
2
0
x2
4 " x2
dx
a. 1
b. 2
c. =
4
d. =
e. 2=
11. Compute
a.
b.
c.
d.
e.
12.
;
.
1
;
=/2
sin 6 x cos 3 x dx
0
1
63
2
63
1
21
4
63
2
21
1
dx
x e 2x
is
a. Convergent by comparison to
b. Divergent by comparison to
;
;
;
;
.
1
c. Convergent by comparison to
d. Divergent by comparison to
.
1
.
1 dx
e 2x
1 dx
e. 2x
1 dx
x
1
1
1 dx
x
4
13.
The basket shown at the right is 9 in tall.
Its horizontal cross sections are circles
whose radius is given by
r2 y
where y is the height from the bottom.
Find the volume of the basket.
216=
162=
108=
81=
e. 81=
2
a.
b.
c.
d.
14.
A trough filled with water is 3m long and its
end is a 45° right triangle which is 2m high
and 2m wide. Find the work done to pump
the water out of the top.
(> is the density of water and
g is the acceleration of gravity.)
a.
b.
c.
d.
e.
2>g
3>g
4>g
6>g
8>g
5
15. (10 points) Compute
;
1
x arctan x dx.
0
6
16. (10 points) Solve the differential equation
condition
yŸ1 ln 4.
x2
2
dy
xy 2x 2
dx
1x
with the initial
7
17. (10 points) Approximate
;
0.1
e "x dx
2
to 7 decimal places.
0
(4 points extra credit:) How do you know it is accurate to 7 decimal places?
8