MATH 152
Final Exam
Sections 510,511
Fall 2000
Solutions
1-12
/48
13
/13
14
/13
15
/13
16
/13
P. Yasskin
Multiple Choice: (4 points each)
1. Compute
2
lim
nv. n!
correctchoice
a. 0
b.
c.
d.
e.
n1
1
2
4
.
n1
n1
2
2 2 2 2 C 2 2 lim 2 2
nlim
lim
nv. 1
nv. n!
v. 1 2 3 C n
2
2
since all but the first 2 factors are less than .
3
2. Compute
;
=/4
0
2
3
C 2 2 lim 2
n
nv. 1
2
2
2
3
n"2
0
tan 2 sec 2 2 d2
a. " 1
2
b. " 1
3
c. 0
d. 1
3
e. 1
2
correctchoice
u tan 2
du sec 2 2 d2
;
=/4
0
tan 2 sec 2 2 d2 ;
1
2
u du u
2
0
1
0
1
2
3. The region below y 1
x
above the x-axis between x 1 and x 2 is rotated
about the y-axis. Find the volume of the solid of revolution.
a. 4=
b. 2=
correctchoice
c. =
d. =
2
e. =
4
Use x-integral and cylinders.
V
;
2=rh dx ;
2
1
2=x 1x dx 2=
1
4. Compute
a.
b.
c.
d.
e.
;
1/2
0
;
1/2
0
x cosŸ=x dx
1 " 1
2=
=2
1 1
2=
=2
2
= " 2=
"2= " = 2
1 " 1
2=
=2
correctchoice
x cosŸ=x dx x
sinŸ=x =
1/2
0
sinŸ=x cosŸ=x x =
=2
"
1/2
0
;
1/2
0
ux
sinŸ=x dx
=
1
2
sin =
2
=
du dx
cos =
2
2
=
" 0
dv cosŸ=x dx
sinŸ=x v
=
cosŸ0 =2
1 " 12
2=
=
5. Find the volume of the parallelepiped whose adjacent edges are
u Ÿ2, "1, 3 , v Ÿ0, 1, 2 and w
Ÿ3, 0, 1 .
a. 1
b. "1
c. 13
correctchoice
d. "13
e. 10 7
| det
v • w
V |u
2 "1 3
0
1 2
3
0 1
|"13| 13
x
2
lim e " 14 " x
xv0
x
2
6. Compute
a. " 1
6
b. " 1
2
c. 0
d. 1
2
e. 1
6
correctchoice
2
3
2
ex 1 x2 ex 1 x x x C
2
6
x4 x6 C
x2
2
6
lim
lim e " 14 " x lim 2
xv0
xv0
xv0
x4
x
x4 x6 C
2
6
1 x2 C
2
6
4
6
2
ex " 1 " x2 x x C
2
6
1
2
2
lb sugar
.
gal water
gal
Pure water is added at the rate of 5
. The mixture is kept well mixed and drained
hr
gal
. How many hours does it take until the concentration drops
at the rate of 5
hr
lb sugar
.
to 0.3
gal water
7. A 50 gal tank is initially filled with salt water whose concentration is 0.6
a. T "10 ln 1
b. T 10 ln 1
correctchoice
2
2
c. T ln 5
d. T " 1 ln 2
10
1
e. T ln 2
10
dS lb " 5 gal S lb " 1 S
®
10
dt hr
hr 50 gal
SŸ0 0.6 lb 50 gal 30 lb
®
gal
SŸT 0.3 lb 50 gal 15 lb
®
gal
®
e "T/10 1
®
T "10 ln
2
;
8. Compute
2
" 2 4/3
15 30e "T/10
1
2
dx
3
e. .
3
1
2 Ÿx
S 30e "t/10
".
"3
"1
a.
b.
c.
d.
;
3
S Ae "t/10
correctchoice
1
dx 4/3
Ÿx " 2 ;
3
2
Ÿ
x " 2 "4/3 dx "3Ÿx " 2 "1/3
3
2
"3 3 lim
xv2
1
.
1/3
Ÿx " 2 9. The curve y sin x between x 0 and x = is rotated about the x-axis.
Which formula will give the surface area of the surface of revolution?
a. A b. A c. A d. A e. A A
;
=
0
;
;
;
;
;
=
0
=
0
=
0
=
0
=
0
2=x 1 cos 2 x dx
1 cos 2 x dx
2= sin x 1 cos 2 x dx
correctchoice
2=x 1 sin 2 x dx
1 sin 2 x dx
2=r ds ;
=
0
2=y 1 dy
dx
2
dx ;
=
0
2= sin x 1 cos 2 x dx
3
10. Find the solution of the differential equation x
condition yŸ2 4.
dy
" 2y 4x 3 satisfying the initial
dx
a. y 4x 3 " 127 x 2
8
b. y 4x 3 " 28
c. y 4x 3 " 7x 2
correctchoice
d. y 4 x 3 " 12
5
5
48
4
3
e. y x "
5
5x 2
dy
" 2x y 4x 2
dx
; P dx
e "2 ln x x "2 12
P " 2x
Pdx " 2x dx "2 ln x
®
Ie
®
x
y
y
1 dy " 2 y 4
d
4
4x C
®
®
dx x 2
x 2 dx
x3
x2
4 4Ÿ2 C
®
®
Initial Condition:
C "7
x 2, y 4
22
y
®
4x " 7
®
y 4x 3 " 7x 2
x2
Standard form:
;
;
!
.
11. Find the radius of convergence of the series
n1
Ÿ
n 1 2
n
Ÿx " 2 .
3n
a. 0
b. 1
2
c. 2
d. 1
3
e. 3
correctchoice
a n1 lim Ÿn 2 2 Ÿx " 2 n1
3n
L nlim
n1
n
2
nv.
v. a n
3
Ÿn 1 Ÿx " 2 |x " 2|
1
®
|x " 2| 3
®
R3
L
3
|x " 2|
lim
3 nv.
n 2 2
2
Ÿn 1 Ÿ
12. Find an equation of the plane perpendicular to the line
x 2 " 2t y "1 3t z 3 t
which contains the point P Ÿ1, 2, 3 a. 2x " y 3z 9
b. "2x 3y z 7
correctchoice
c. x 2y 3z 14
d. 2x y z 7
e. 2x 2y z 9
v Ÿ"2, 3, 1 .
The normal to the plane is the tangent to the line: N
So the equation of the plane is
ŸX " P 0 ® Ÿ"2, 3, 1 Ÿx " 1, y " 2, z " 3 0 ® " 2x 3y z " 7 0
N
4
Work Out (13 points each)
;
Show all your work. Partial credit will be given. You may not use a calculator.
32
13. Compute
dx
2
Ÿx " 2 Ÿx 2 Ÿx 4 32
A B Cx2 D
x"2
x2
x 4
x " 2 Ÿx 2 Ÿx 2 4 2
2
32 AŸx 2 Ÿx 4 BŸx " 2 Ÿx 4 ŸCx D Ÿx 2 " 4 x 2:
32 AŸ4 Ÿ8 ®
A1
x "2:
32 BŸ"4 Ÿ8 ®
B "1
x 0:
32 AŸ2 Ÿ4 BŸ"2 Ÿ4 DŸ"4 16 " 4D
®
D "4
x 1:
32 AŸ3 Ÿ5 BŸ"1 Ÿ5 ŸC D Ÿ"3 20 " 3ŸC " 4 ®
C0
32
1
"
1
"
4
2
2
x"2
x2
x 4
Ÿx " 2 Ÿx 2 Ÿx 4 32
1 "1 "4 dx
dx x 2 tan 2
2
x"2
x2
x2 4
Ÿx " 2 Ÿx 2 Ÿx 4 2 sec 2 2 d2 lnŸx " 2 " lnŸx 2 " 2 d2
lnŸx " 2 " lnŸx 2 " 4
4 tan 2 2 4
lnŸx " 2 " lnŸx 2 " 22 C lnŸx " 2 " lnŸx 2 " 2 tan "1 x C
2
Ÿ
;
14.
;
;
;
A water tank has the shape of a cone with the vertex
at the top. The cone is 8 m tall and has a base which
is a circle of radius 4 m. The cone is filled with water
to a depth of 5 m. How much work is needed to pump
the water out the top of the tank?
kg
and
m3
the acceleration of gravity is g 9.8 m 2 ,
sec
but you may leave your answer as a multiple of >g.)
(The density of water is > 1000
Measure y down from the top. There is water where 3 t y t 8
and the water at height y must be lifted a distance D y.
The water at height y is a circular disk of thickness dy and
radius r satisfying yr 4 or r 1 y. So its volume is
8
2
2
=
y
dy and its weight is dF >gdV.
dV =r 2 dy 4
Therefore the work is
W
;
D dF ;
8
3
y >g
=y 2
dy >g =
4
4
;
y 3 dy >g =
4
3
8
y4
4
8
3
>g = Ÿ8 4 " 3 4 16
5
15. Determine if each of the following series converges or diverges. Say why.
Be sure to name or quote the test(s) you use and check out all requirements of the test.
If it converges, find the sum. If it diverges, does it diverge to ., ". or neither?
!
.
a.
1
lnn
n
n2
Explain:
Circle one: Converges
Apply Integral Test:
1
is positive and decreasing
Let u lnn
n lnn
.
1 dn 1 du 2 u 2 lnn . .
2
2 n lnn
u
;
!
.
So
n2
!
.
b.
Ÿ
;
1
n lnn
"1 n 3
du 1n dn
diverges to .
n
Circle one:
n!
n0
Diverges
Converges
Diverges
Explain:
Apply Alternating Series Test:
(Unnecessary if you get the sum.)
n
n
3
3 n 0.
is positive and decreasing for n 3.
lim
Ÿ"1 is alternating.
nv. n!
n!
.
n
n 3
So
converges.
Ÿ" 1 n!
n
!
0
OR
ex !
!
.
n0
.
So
Ÿ
n0
xn
n!
which converges for all x.
"1 n 3
n
n!
e "3
16. Find the arclength of the curve x y3
1
4y
3
for 1 t y t 3.
dx y 2 " 1
dy
4y 2
2
1 y 4 " 2y 2 1 2 1 4 y 4 1 1 4 y 2 1 2
1 dx
2
dy
16y
16y
4y
4y
3
2
3
3
3
y
9" 1
1 dx
" 1
L
dy y 2 1 2 dy 12
3
4y 1
dy
4y
1
1
;
;
2
"
1 " 1
3
4
53
6
6