CALCULUS TWO
TEST TWO
NAME (In ink.):
So L
" T Iu/VS
SHOW ALL CALCULATIONS. GIVE MATHEMATICALLY EXACT FORM OF ANSWERS. OCT. 23,2008.
BEST TEN 10 PT. PROBLEMS COUNT, FOR A SCORE OUT OF 100.
Page 1 of 6.
1. Solve the initial value problem
2. Solve the differential equation
y' = eX(y2 + 1), y = 1 if x = ln(n16)
xy' - 5y = x6 sec2x
CALCULUS TWO
TEST TWO
So~ur~o/\/S
NAME (In ink.):
SHOW ALL CALCULATIONS. GIVE MATHEMATICALLY EXACT FORM OF ANSWERS. OCT. 23,2008.
BEST TEN 10 PT. PROBLEMS COUNT, FOR A SCORE OUT OF 100.
Page 2 of 6.
r = 6 COS 6
3. Given the polar equation
+
10 sin 6
(a) Convert it to an equation in rectangular coordinates.
r
= G V - @+
10
h&t&
1~
P&B
\ x ~ * ~ ' 6%
= +IU3
*
Describe the graph in detail.
Sketch the graph.
--
4. A curve is defined parametrically by
x = ,,/
y=
= Jf, 0 5 t
< 25.
1
(a) Eliminate the parameter and i
(b) Find
$=
dt
dy
as a function of the parameter, t.
I zfi;-t3/
(c) Set up a simplified integral to find the arclength of the curve.
GO TO PAGE 3.
CALCULUS TWO
TEST TWO
NAME (In ink.):
SO L M 7
1 0NS
SHOW ALL CALCULATIONS. GIVE MATHEMATICALLY EXACT FORM OF ANSWERS. OCT. 23,2008.
Page 3 of 6.
BEST TEN 10 PT. PROBLEMS COUNT, FOR A SCORE OUT OF 100.
5. Consider the rose
r = 8 cos(20) and the circle r
(a) Graph them together, giving polar
(Can use polar graph paper.
=
4fi .
rdinates ot the points ot ~ntersecdon.
-
/
tn
-
13-/
12-
'-
(b) Find the area of the region outside the circle, but inside the rose.
TO PAGE 4.
- --
2 U +711
IS/
~118
IZ
CALCULUS TWO
TEST TWO
NAME (In ink.):
SOL
I OMS
SHOW ALL CALCULATIONS. GIVE MATHEMATICALLY EXACT FORM OF ANSWERS. OCT. 23,2008.
Page 4 of 6.
BEST TEN 10 PT. PROBLEMS COUNT, FOR A SCORE OUT OF 100.
6. Find the arclength of the curve
x3
y = 24
+ y2
from x = 1 to x = 3.
%.
7. Find the area of the surface of revolution formed by revolving the portion of the graph of
x2 - 3y = 3 from (0,- 1) to (2,
about the y-axis.
i),
?I
GO TO PAGE 5.
x2-3
=?3,
J"
:X
CALCULUS TWO
TEST TWO
NAME (In ink.):
SOLHTIO~S
SHOW ALL CALCULATIONS. GIVE MATHEMATICALLY EXACT FORM OF ANSWERS. OCT. 23,2008.
BEST TEN 10 PT. PROBLEMS COUNT, FOR A SCORE OUT OF 100.
Page 5 of 6.
8. Find the centroid of the quarter disk in the first quadrant bounded by a circle of radius 5
centered at the origin. Draw an appropriate diagram. with coordinates.
-
-
-
F =
5
A
c
123-
- . -3
1
L
f
3
3
tf7i
-
"
F
id
C b + o
-'
0
4-
l 2 r
20
37T
2o
i s
L
9.
4-
7
2-jF
1
38
-1
Find the slopes of the tangent lines to the curve given parametrically by
r = 2 - Ssin t at the point of self intersection.
Sketch the graph and the tangent lines.
S,\F
ib+.rr,+:~
ir
L
r+
/&.
=rmt =
2
az
TCt=&-'e)
t
3
3
-
GO TO PAGE 6.
_--
--
-
I
=
s
CALCULUS TWO
TEST TWO
NAME (In ink.):
SQL M 7 1 0 d S
SHOW ALL CALCULATIONS. GIVE MATHEMATICALLY EXACT FORM OF ANSWERS. OCT. 23,2008.
Page 6 of 6.
BEST TEN 10 PT. PROBLEMS COUNT, FOR A SCORE OUT OF 100.
10. A plate in the form of a right triangle with base 5 meters and height 10 meters is submerged
vertically in water, with its base up, parallet to the surface of the water, and 3 meters below
the surface. Find the force on one side of the plate. Assume pg = 9800.
11. Solve the differential equation.
I-h
u=r
- 1-2- - I
-3 - 3 ,
(100 points, total.)
y'
+
3~
- = x3y2sin x. Hint: Bernoulli substitution.
X
,J1=-3 3 .