Name
ID
MATH 172
FINAL EXAM
Section 502
Solutions
Fall 1998
P. Yasskin
Multiple Choice: (8 points each)
;
1. Compute
a. " 1
=
0
x sinŸx 2 dx
2
b. 0
c. 1
2
d. 1
e. 2
;
=
correctchoice
x sinŸx 2 dx
0
;
2. Compute
"3e
"3e 2
"3e " 2
a.
b.
c.
d.
e
e. e " 2
u
2x dx
u
x
" cosŸ= 2
0
v
dv
dx
e
x
e x dx
v
;
e x dx
e
1
x 2 e x dx
" "1 " "1
2
2
0
x
;
;
x 2 e x " 2 x e x dx
1
1
0
1
x 2 e x " 2 xe x " e x dx
1
x 2 e x " 2Ÿxe x " e x 3. Find the average value of the function
a. 13
fŸx ¡e
0
3x 2
0
" 2Ÿe " e ¢ " ¡0 " 2Ÿ0 " 1 ¢
for
1
e"2
14
1 t x t 3.
correctchoice
b. 14
c. 15
d. 16
e. 17
2
x 2 e x dx
f ave
" cosŸ0 "
0
dv
du
du
2
correctchoice
x2
1
=
" cosŸx 2 1
b"a
;
b
a
fŸx dx
1
2
;
3
1
2
Ÿ3x 1 dx
1 Ÿx 3
2
x 3
1
27 3
2
"
11
2
1
;
x2
dx
2 3/2
Ÿ1 " x Hints:
sin 2 2 cos 2 2 1
tan 2 2 1 sec 2 2
x
a.
" arctan x
1 " x2
x
arctan x
b.
1 " x2
x
correctchoice
c.
" arcsin x
1 " x2
x
arcsin x
d.
1 " x2
x
x
e.
1 " x2
4. Compute
Trig sub:
x sin 2
dx cos 2 d2
cos 2 1 " x 2
2
x2
dx sin 32 cos 2 d2 tan 2 2 d2 sec 2 2 " 1 d2
2 3/2
cos 2
Ÿ1 " x x
" arcsin x
1 " x2
;
;
;
5. Find the area between the curves
a.
b.
c.
d.
e.
A
1
24
1
12
1
7
1
6
1
;
1
0
Ÿx
y
;
x2
and
y
x3
2
" x 3 dx
x3 " x4
3
4
1
0
1 " 1
3
4
;
0 t x t 1.
1
12
and
y x2
y x3
rotated around the x-axis. Find the volume swept out.
a. =
5
b. =
10
c. =
20
correctchoice
d. 2=
35
e. 4=
35
Washers:
for
" 2¢
correctchoice
6. The area between the curves
V
¡tan 2
=R 2 " =r 2 dx
;
1
0
=Ÿx 2 " =Ÿx 3 dx
2
2
=x 5 " =x 7
5
7
for
1
0
0txt1
= " =
7
5
is
2=
35
2
bar whose linear density is
7. Find the total mass of a
6 cm
> Ÿ1 x g/cm
where x is the distance from one end in cm.
a. 7 g
6
b. 4 g
c. 6 g
d. 7 g
e. 24 g
M
;
> dx
correctchoice
;
6
0
Ÿ1 x dx
2
x x
2
6
0
6 36
2
24
8. Which of the following is the direction field for the differential equation
a.
No, vert at y
0
d
b.
No,all pos slope
e.
c.
No,all pos slope
dy
dx
x2 ?
y2
No, hor at x
0
correctchoice
3
;
.
1
dx
1 x2
correctchoice
9. Compute
a. =
4
=
b.
1
2
3
c. =
4
d. Convergent but none of the above
e. Divergent
;
.
1
1
1x
2
dx
!
.
10. Compute
n1
a. =
4
b. =
.
arctan x
arctan . " arctan 1
1
1
1 n2
= " =
4
2
=
4
2
c. 3=
4
d. Convergent but none of the above
e. Divergent
;
;
Since
Since
.
1
1
1 x2
.
0
1
1 x2
!
.
11. The series
dx
dx
Ÿ" 1 n1
correctchoice
is convergent, so is
=
2
=
4
n
!
.
=
4
also,
n
1 n2
!
.
n1
1 .
1 n2
n1
1
1 n2
=
2
is
a. absolutely convergent
b. conditionally convergent
correctchoice
c. divergent
d. none of these
!
.
Ÿ" 1 n
n1
lim
nv.
n
1 n2
because
n
1 n2
0
;
.
1
is convergent because it is an alternating, decreasing series and
!
.
However, the related absolute series
x
1 x2
dx
1 lnŸ1 x 2 2
.
1
..
n1
n
1 n2
is divergent
4
!
.
12. Find the radius of convergence of the series
Ÿx
n2
" 3 n .
2nn2
a. 0
b. 1
correctchoice
c. 2
d. 3
e. 4
an
Ÿx
" 3 n
a n1
2nn2
n1
Ÿx " 3 2nn2
lim
nv. 2 n1 n 1 2 Ÿx " 3 n
Ÿ
a Ÿ2,
13. The vectors
Ÿx " 3 2 n1 Ÿn n1
1 2
|x " 3|
lim
2 nv.
"1, 4 and
|x " 3|
2
"1 are
n2
2
Ÿn 1 b
Ÿ3, 2,
1
|x " 3|
2
R
a. parallel
b. perpendicular
correctchoice
c. neither
a
b
6"2"4
0
14. Find the area of the parallelogram whose edges are
a Ÿ1, 2, 3 and
Ÿ3, 2, 1 .
b
2
4
4
2
e. 4
a.
b.
c.
d.
2
2
3
6
6
correctchoice
î F k
a•
b
1 2 3
îŸ2
" 6 " F Ÿ1 " 9 k Ÿ2 " 6 Ÿ
"4, 8, "4 3 2 1
A
a•
b
16 64 16
96
4 6
15. A baseball is thrown straight North initially at 45 ( above horizontal and follows a
parabolic path, up and back down. At the top of the trajectory, in what direction does
the unit binormal
B
point?
a. West and horizontal
b.
c.
d.
e.
correctchoice
West and below horizontal
Straight down
East and below horizontal
East and horizontal
is straight down.
T is North and horizontal.
N
So by the right hand rule, B is West and horizontal.
5
Work-out Problems: (20 points each)
dy
dx
16. Solve the differential equation
yŸ1 2xy
e "x
2
with the initial condition
0.
; Pdx
; 2xdx
x
e
e
Linear:
Ie
P 2x
Q e "x
U
2 dy
2
2
x2
x 2 "x 2
e 2xy e e
e x y 1dx x C
ex
ex y 1
dx
2
x 1 when y 0 :
e10 1 C
So
C "1
ex y x " 1
2
17. Find the point where the line
2x " 3y z
2
;
x
"2 t
y
1 2t
z
3 " 2t
Ÿx
" 1 e "x
2
intersects the plane
"16.
Plug the line into the plane and solve for t:
2Ÿ"2 t " 3Ÿ1 2t Ÿ3 " 2t "16
" 4 " 6t "16
Plug back into the line:
x "2 t "2 2 0
y 1 2t 1 4 5
So the point is
Ÿx, y, z Ÿ0, 5, "1 " 6t
z
The spiral at the right is made from an infinite
18.
y
3 " 2t
"12
0.5
3"4
1
t
"1
1.5
2
number of semicircles whose centers are all
on the x-axis. The radius of each semicircle
is half of the radius of the previous semicircle.
a. Consider the infinite sequence of points where the spiral crosses the x-axis.
What is the x-coordinate of the limit of this sequence?
2"1 1 " 1
4
2
1 "C
8
!
.
n0
n
2 "1
2
2
1
1
2
4
3
b. What is the total length of the spiral (with an infinite number of semicircles)?
Or, is the length infinite?
Each semicircle has length
Ln
!
=r n
.
So the total length is
L
n0
!
.
=r n
n0
where the radii are r n
= 1
2
n
=
1" 1
2
2
1, 1 , 1 , 1 , C.
2 4 8
2=.
6
rŸt Ÿ6t, 3t 2 , t 3 19. Consider the twisted cubic curve
a. Find the arc length of the curve between
v Ÿ6, 6t, 3t 2 L
; ;
ds
|v|dt
|v|
36 36t 2
;
2
0
Ÿ6 3t 2 dt
9t 4
6t t 3
t
Ÿ6 0
3t 2 2
for
0 t t t 2.
and
t 2.
6 3t 2
2
12 8
20
0
b. Find the unit binormal vector
B .
a Ÿ0, 6, 6t F
î
v•
a
6 6t 3t 2
0 6
Ÿ18t
v•
a
B
k
2
îŸ36t
2
" 18t 2 " F Ÿ36t " 0 k Ÿ36 " 0 6t
, "36t, 36 18Ÿt 2 , "2t, 2 18 t 4 4t 2 4 18 Ÿt 2 2 2 18Ÿt 2 2 v•
a 18
t 2 , "2t , 2
2
Ÿt , "2t, 2 2
2
18Ÿt 2 t 2 t2 2 t2 2
v•
a
7