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MATH 152
Final Exam
Sections 201,202
1. Compute
a.
b.
c.
d.
e.
b.
c.
d.
e.
13
/13
14
/13
15
/13
16
/13
2n
e "2
1
e 1/2
e2
.
2. Compute
a.
1 " 1n
lim
nv.
/48
Spring 2001
P. Yasskin
Multiple Choice: (4 points each)
1-12
;
=/4
0
sin 2 2 cos 2 2 d2
=
32
=
16
=
8
=
4
=
2
3. The region below y 1
x
above the x-axis between x 1 and x . is rotated
about the x-axis. Find the volume of the solid of revolution.
a. =
4
b. =
2
c. =
d. 2=
e. 4=
1
4. Compute
a.
b.
c.
d.
e.
;
1
x 2 e "x dx
0
"5e "1
2 " 5e "1
"e "1
2 " e "1
e "1 " 2
Ÿ Ÿx
5. Find the average value of the function f x a.
b.
c.
d.
e.
1
9
2
3/2
on the interval
¡0, 3 ¢.
2
6
2
27
2
9
1
9 2
1
27 2
Ÿ
6. Find the angle between the vector u 2, 1, "2
Ÿ
P 3, "4, 12
Ÿ
containing the vectors v 1, 0, 0
and the normal to the plane through
0, "3, 4 .
and w
Ÿ
a. arccos "22
b. arccos
c. arccos
d. arccos
e. arccos
39
15
2
3
2
2
15
2
3
2
7. Use the 4 th degree Maclaurin polynomial for e "x
2
to estimate
;
1
e "x dx.
2
0
a. 1 " 1 1
3
5
b. 1 1 1
5
3
1
1
c. 1 "
120
6
1
1
d. 1 10
3
e. 1 " 1 1
10
3
!
.
8. The series
n1
n
n 3/2 1
is
!
.
a. convergent by the Comparison Test with
!
.
b. conv. by the Limit Comp. Test with
n1
c. divergent by the Comparison Test with
!
.
d. div. by the Limit Comp. Test with
n1
n1
1
n 1/2
1 but not by the Comp. Test
n 1/2
!
.
n1
1
n 1/2
1 but not by the Comp. Test
n 1/2
e. none of these
9. The area below y x 2 , 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. 15=
4
15
=
c.
2
d. 8=
e. 31=
5
3
!
.
10. Compute
n1 " n2
n
n"1
n2
a.
b.
c.
d.
e.
0
1
2
3
divergent
11. The Maclaurin series for sinh x is
sinh x !Ÿ
.
k0
3
5
7
x 2k1
x x x x 2k 1 !
3!
5!
7!
C
If you use the 5 th -degree Maclaurin polynomial to approximate sinh x on the interval
1 , 2 , bound the error in the approximation using the Taylor Remainder Inequality
2
M |x " a|n1
where M u |f Ÿn1 c | for all c between x and a.
|R n x | t
n1 !
Ÿ Ÿ
Ÿ HINT: sinh x:
cosh x:
-2
a.
b.
c.
d.
e.
4
15
4
45
4
45
4
15
4
45
-1
1x
2
-2
-1
1x
2
cosh 2
sinh 2
cosh 1
2
1
sinh
2
cosh 2
Ÿa, b, c where the line x 2 " t y 3 2t z 4 t
intersects the plane 2x " y 3z 14. Then a b c 12. Find the point
a.
b.
c.
d.
e.
1
3
5
7
9
4
Work Out (13 points each)
Show all your work. Partial credit will be given. You may not use a calculator.
dy
13. Find the solution of the differential equation x 3
" 2y 4 satisfying the initial
dx
condition y 1 3.
Ÿ 14. The curve y x 2 for 0 t x t
2
is rotated about the y-axis. Find the surface area
of the resulting surface.
5
15.
A plate in the shape of a semicircle is placed at
the bottom if a tank with the straight edge down.
The radius of the circle is 4 cm and the water
in the tank is 6 cm deep.
What is the force on the plate?
(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 in terms of >g.)
6
! ŸŸ
.
16. Find the interval of convergence of the series
n2
x"3
n lnn
n
2
.
Be sure to check the endpoints.
Name or quote the test(s) you use and check out all requirements of the test.
7