Name
Section
MATH 152
FINAL EXAM
Spring 2014
Sections 549-551
1-14
/56
15
/15
16
/20
17
/10
18
/5
P. Yasskin
Multiple Choice: (14 problems, 4 points each)
Total
1.
Find the area between the curves y
a.
b.
c.
d.
e.
2.
2x 2 and y
b.
c.
d.
e.
4x.
4
3
8
3
16
3
32
3
64
3
Find the average value of the function f x
a.
/106
e 1e
1 e
2
e 1e
1 e
2
1
e x on the interval
1, 1 .
1
e
1
e
1
3.
a.
b.
c.
d.
e.
4.
/2
b.
c.
d.
e.
x 2 e 2x dx.
1 x2
2
1 x2
2
1 x2
2
1 x2
2
1 x2
2
1x
2
1x
2
1x
2
1x
2
1x
2
b.
c.
d.
e.
arcsec x
x2
1
4
1
4
1
4
1
2
1
2
e 2x
C
e 2x
C
e 2x
C
e 2x
C
e 2x
C
x2 1
dx.
x
Compute
a.
sin 4 cos d .
6
2
5
2
5
2
3
6
Compute
a.
5.
/2
Compute
1
x2
1
C
arcsec x
C
x 2 1 arcsec x C
1 x 2 1 3/2 x 2 1
3
1 x 2 1 3/2 x 2 1
3
HINT: tan 2
1/2
C
1/2
C
sec 2
1
2
6.
In the partial fraction expansion,
c.
1
4
1
2
2
d.
8
e.
16
a.
b.
7.
Compute
2
b.
c.
d.
e.
8.
A
x
B
x2
C
x3
Dx E
x2 4
Fx G , find C.
2
x2 4
1 dx.
x3
2
a.
4x 5 2x 2 8
2
x3 x2 4
1
4
1
8
0
1
4
undefined
The region under the curve y
1
above the interval
x 1
y-axis. Find the volume of the resulting solid.
a.
ln 2
b.
ln 2
c.
2 ln 2
d.
4 ln 2
e.
4 ln 2
2
0, 1
is revolved about the
4
3
9.
Find the arclength of the curve x
a.
b.
c.
d.
e.
2t 4 ,
y
t 6 between t
0 and t
1.
1
27
1
3
61
12
61
27
61
54
2t 4 , y t 6 between t 0 and t 1 is rotated about the x-axis.
Which integral gives the area of the resulting surface?
10. The curve x
1
a.
0
1
b.
0
1
c.
0
1
d.
0
1
e.
4 t 9 16
9t 4 dt
8 t 7 16
9t 4 dt
4 t 7 16
9t 4 dt
2 t 7 16
9t 4 dt
4 t 3 16
9t 4 dt
0
11. The series S
n 0
a.
b.
c.
d.
e.
1 n3n
2 2n
converges to 4
converges to 4
3
converges to 4
7
converges to 2
5
diverges
4
x 3 6 sin x
x5
12. Compute lim 6x
x 0
a.
b.
c.
d.
e.
6
5!
6
6
2
3!
1
20
1
13. Compute
n 0
b.
1
2
1
c.
0
d.
1
a.
e.
n
2n
9 n 2n !
1
2
14. Find the center and radius of the sphere x 2
a.
center:
2, 0, 3
radius: R
2
b.
center: 2, 0, 3
radius: R
9
c.
center:
2, 0, 3
radius: R
9
d.
center: 2, 0, 3
radius: R
3
e.
center:
radius: R
3
2, 0, 3
4x
y2
z2
6z
4
0
5
Work Out (4 questions, Points indicated)
Show all you work.
15.
15 points
Consider the series S
n 1
a.
b.
c.
d.
3
1 n 1n
.
2n 1
Show the series is convergent.
6 Extra Credit
Show the series is absolutely convergent.
HINT: You may use these formulas without proof, if neeeded:
bx
2 n 1 n 3 for n 12
b x dx
C
x b x dx
ln b
3
xb x
ln b
bx
ln 2 b
C
Compute S 7 , the 7 th partial sum for S. Do not simplify.
3 Find a bound on the remainder |R 7 |
S. Name the theorem you used.
|S
S 7 | when S 7 is used to approximate
6
16.
20 points
Let f x
ln x .
a.
6 Find the Taylor series for f x
b.
centered at x
11 The Taylor series for f x centered at x
3.
4 is T x
2 ln 2
n 1
Find the interval of convergence for the Taylor series centered at x
radius and check the endpoints.
c.
1 n
4nn
1
x
4 n.
4. Give the
3 If the cubic Taylor polynomial centered at x 4 is used to approximate ln x
on the interval 3, 6 , use the Taylor’s Inequality to bound the error.
Taylor’s Inequality:
Let T n x be the n th -degree Taylor polynomial for f x centered at x a
and let R n x
f x T n x be the remainder. Then
M |x a| n 1
|R n x |
n 1 !
provided M |f n 1 c | for all c between a and x.
7
17.
10 points
y
A conical tank with vertex down, radius 4,
8
6
and height 8, is filled with water.
Find the work done to pump the water out the top of the tank.
4
Take the density of water to be
2
and the acceleration of gravity to be g.
-4
18.
5 points
The region between the curves y
Find the volume of the resulting solid.
2x 2 and y
-2
0
2
4
4x is rotated about the x-axis.
8