Name
ID
MATH 253
FINAL EXAM
Sections 501-503
Section
Fall 1998
1-12
/120
13
/25
14
/15
15
/30
16
/20
P. Yasskin
Multiple Choice: (10 points each)
1 1 x x2 x3 C
1"x
2
3
ex 1 x x x C
2
3!
HINTS:
1. Compute
a.
b.
c.
d.
e.
3
5
7
sin x x " x x " x C
3!
5!
7!
2
4
6
x
x
cos x 1 "
" x C
2!
4!
6!
3n 2
lim
nv. 1 n 3
0
1
2
3
Divergent
2. Find
r
5 5r 5r 2 5r 3 5r 4 C 3.
such that
a. 2
b.
c.
d.
e.
5
"2
5
3
5
5
3
"2
3
!
99
3. Compute
k1
a.
b.
c.
d.
e.
1
k
"
1
k1
.9
.99
1
1.1
Divergent
1
!
.
4. The series
n1
1
n n
2
is
!
!
.
a. divergent by comparison to
n1
b. convergent by comparison to
1 .
n
.
n1
1 .
n2
c. divergent by the ratio test.
d. convergent by the ratio test.
e. divergent by the n th -term test.
!
.
5. The series
Ÿ" 1 n1
a.
b.
c.
d.
e.
n
3n 2
1 n3
is
absolutely convergent.
conditionally convergent.
divergent to ..
divergent to "..
oscillatory divergent.
6. Compute
lim
xv0
cosŸ2x " 1 2x 2
x4
a. 0
1
24
c. 1
12
d. 2
3
e. .
b.
!
.
7. Given that
n0
a.
b.
c.
d.
e.
x 1
1"x
n
(for |x| 1),
then (for |x| 1) we have
!
.
n xn n0
1
1"x
1
2
Ÿ1 " x x
2
Ÿ1 " x x
1"x
n
1"x
2
8. Find the x-coordinate of the center of mass of the rectangle
0tyt2
if the density is
> xy.
a. .5
b. 1
c. 1.5
d. 2
e. 2.5
9. Find the mass of the solid inside the cylinder
z x y
a. =
2
b. 2=
3
c. =
d. 3=
2
e. 2=
2
10. If
a.
b.
c.
d.
e.
11. If
a.
b.
c.
d.
e.
2
and below the plane
z2
F Ÿxz 2 , "yz 2 , z 3 2 2
2
Ÿz , z , 3z 2
2
2
Ÿz , "z , 3z 3z 2
5z 2
2Ÿy " x z
then
F
F Ÿxz 2 , "yz 2 , z 3 2Ÿy " x z
2Ÿx y z
Ÿ2yz, "2xz, 0 Ÿ2yz, 2xz, 0 0
then
•F
0 t x t 3,
above the paraboloid
x2 y2 1
if the density is
> x2 y2.
3
12. Compute
from
;
t=
rŸt y 2 e xy dx Ÿ1 xy e xy dy
to
t 3=.
along the spiral
rŸt Ÿt cos t, t sin t Ÿy 2 e xy , Ÿ1 xy e xy .
F
HINT:
Find a scalar potential for
a. "2=
b. 0
c. =
d. 3=
e. 4=
Work-Out Problems
13. (25 points) You are given:
a. (10 pt) If
fŸx e
"x 2
e
"x 2
!
.
Ÿ" 1 n
n0
,
;
find
x 2n 1 " x 2 x 4 " x 6 C.
2
6
n!
f Ÿ6 Ÿ0 .
b. (10 pt) Use the quadratic Taylor polynomial approximation about
e
"x 2
to estimate
0.1
e
"x 2
x0
for
(Keep 8 digits.)
dx.
0
c. (5 pt) Your result in (b) is equal to
;
0.1
e
0
"x 2
dx
to within o how much? Why?
4
!
.
14. (15 points) Find the interval of convergence for the series
n0
" 5 n
.
3nn3
Ÿx
Be sure to identify each of the following and give reasons:
(1 pt) Center of Convergence:
a
Radius of Convergence:
(1 pt) Right Endpoint:
R
(5 pt)
x
At the Right Endpoint the Series
Converges
(circle one)
(3 pt)
(circle one)
(3 pt)
Diverges
(1 pt) Left Endpoint:
x
At the Left Endpoint the Series
Converges
Diverges
(1 pt) Interval of Convergence:
5
15. (30 points) Stokes’ Theorem states that if S is a surface in 3-space and S is its
boundary curve traversed counterclockwise as seen from the tip of the normal to S
then
;;
•F
dS >
ds
F
S
S
F Ÿ"yz, xz, z and S is the part of the cone
Verify Stokes’ Theorem if
2
2
z x y
below
z2
with normal pointing in and up.
2
a. (5 pt) Compute
b. (10 pt) Compute
•F
.
;;
(HINT: Use rectangular coordinates.)
•F
dS.
S
(HINT: Here is the parametrization of the cone and the steps you should use.
Remember to check the orientation of the surface.)
Ÿr, 2 Ÿr cos 2, r sin 2, r R
r R
2 R
N
•F
;;
Ÿr, 2 R
•F
dS S
6
>
15c. (15 pt) Compute
ds .
F
Recall
F Ÿ"yz, xz, z 2 .
S
(HINT: Parametrize the boundary circle. Remember to check the orientation of the
curve.)
rŸ2 vŸ2 rŸ2 F
>
ds F
S
16. (20 points) Find the minimum value and its location(s) for the function
on the ellipse
9x 4y 72.
2
2
fŸx, y xy
7