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AP
Calculus
BC
2005 Free-Response
Questions
?
?
?
?
?
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2005 AP
?
CALCULUS
BC FREE-RESPONSE QUESTIONS
CALCULUS BC
SECTION II, Part A
Time
—
45
minutes
Number of
problems
—
3
A graphing calculator is required for
some problems or parts of problems.
1
-
x
1.
Let
f
and
g
be the functions given by
f
(
x
)
=
4
+
sin
(
p
x
)
and
g
(
x
)
=
4
. Let
R
be the
shaded region in
the first quadrant
enclosed by the
y
-axis and
the graphs of
f
and
g
, and let
S
be the shaded region in
the first
quadrant enclosed by the graphs of
f
and
g
, as shown in the figure
above.
(a)
Find the area of
R
.
(b)
Find the area
of
S
.
(c)
Find the
volume of the solid generated when
S
is revolved about the
horizontal line
y
= -
1.
WRITE ALL WORK
IN THE TEST BOOKLET.
Copyright ? 2005 by College Entrance
Examination Board. All rights reserved.
Visit (for AP professionals) and
/apstudents (for AP students and
parents).
GO ON
TO THE NEXT PAGE.
2
2005 AP
?
CALCULUS
BC FREE-RESPONSE QUESTIONS
2. The curve above is drawn in the
xy
-plane and is described by
the equation in polar coordinates
r
=
q
+
sin
(
2
q
)
for
0
?
q
?
p
,
where
r
is measured in
meters and
q
is measured in radians. The derivative
of
r
with respect
to
q
is
dr
given by
d
q
=
1
+
2cos
(
2
q
)
.
(a)
Find the area
bounded by the curve and the
x
-axis.
(b)
Find the
angle
q
that corresponds to
the point on the curve with
x
-coordinate
-
2.
(c)
For
2
p
dr
p
,
is negative.
What does this fact say about
r
? What does this fact say
about the
<
q
<
3
3
d
q
curve?
p
(d)
Find the value of
q
in the interval 0
?
q
?
2
that corresponds to the point on the curve in the
first
quadrant with greatest distance
from the origin. Justify your answer.
WRITE ALL WORK IN THE TEST
BOOKLET.
Copyright ? 2005 by
College Entrance Examination Board. All rights
reserved.
Visit (for AP
professionals) and /apstudents (for AP students
and parents).
GO
ON TO THE NEXT PAGE.
3
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