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MATH 152
Final Exam
Sections 510,511
1. Compute
a.
b.
c.
d.
e.
/48
13
/13
14
/13
15
/13
16
/13
Fall 2000
P. Yasskin
Multiple Choice: (4 points each)
1-12
2 n1
lim
nv. n!
0
1
2
4
.
2. Compute
;
=/4
0
tan 2 sec 2 2 d2
a. " 1
b.
c.
d.
e.
2
"1
3
0
1
3
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.
4=
2=
=
=
2
e. =
4
a.
b.
c.
d.
1
4. Compute
a.
b.
c.
d.
e.
;
1/2
0
x cosŸ=x dx
1 " 1
2=
=2
1 1
2=
=2
2
= " 2=
"2= " = 2
1 " 1
2=
=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.
b.
c.
d.
e.
1
"1
13
"13
10 7
x
2
lim e " 14 " x
xv0
x
2
6. Compute
a. " 1
b.
c.
d.
e.
6
"1
2
0
1
2
1
6
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
2
2
c. T ln 5
d. T " 1 ln 2
10
1
e. T ln 2
10
;
8. Compute
a.
b.
c.
d.
e.
3
2 Ÿx
1
" 2 4/3
dx
".
"3
"1
3
.
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 ;
;
;
;
;
=
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
2=x 1 sin 2 x dx
1 sin 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
d. y 4 x 3 " 12
5
5
48
4
3
e. y x "
5
5x 2
!
.
11. Find the radius of convergence of the series
n1
Ÿn
1 2
3n
Ÿx
" 2 n.
a. 0
b. 1
2
c. 2
d. 1
3
e. 3
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.
b.
c.
d.
e.
2x " y 3z 9
"2x 3y z 7
x 2y 3z 14
2x y z 7
2x 2y z 9
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 5
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?
(The density of water is > 1000 kg3 and
m
the acceleration of gravity is g 9.8 m 2 ,
sec
but you may leave your answer as a multiple of >g.)
6
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:
!
.
b.
Ÿ" 1 n
n0
3n
n!
Circle one: Converges
Diverges
Circle one: Converges
Diverges
Explain:
7
16. Find the arclength of the curve x y3
1
3
4y
for 1 t y t 3.
8