AP Calculus BC - College Board
2018
AP Calculus BC
Sample Student Responses
and Scoring Commentary
Inside:
Free Response Question 5 R Scoring Guideline R? Student Samples R? Scoring Commentary
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AP? CALCULUS BC
2018 SCORING GUIDELINES
Question 5
( ) (a)= Area
1 2
5 3 3
4 2
- (3 + 2cos )2
d
{3 : 1 : constant and limits 2 : integrand
(b)
dr d
= -2sin
dr d =
2
= -2
( ) ( ) r
2
= 3 + 2cos
2
= 3
y =
r sin ddy=
dr d
sin
+
r cos
x =
r cos ddx=
dr
d
cos
-
r sin
( ) ( ) = dy ( ) ( ) dx = 2
dd= yx dd = 2
-2 sin -2 cos
22= +- 33csoins 22
2 3
= 1 : ddy
dr d
sin
+
r cos
= or ddx
dr d
cos
-
r sin
3 :
1
:
dy dx
=
dr d
sin
+
r cos
dr d
cos
-
r sin
1 : answer
The slope of the line tangent to the graph of r= 3 + 2cos
at
=
2
is
2 3
.
-- OR --
y= r sin = (3 + 2cos )sin ddy= 3cos + 2cos2 - 2sin2
x
= r cos
= (3 + 2cos )cos
dx d
= -3sin
- 4sin cos
( ) ( ) ( ) = dy ( ) ( ) ( ) dx = 2
dd= yx dd = 2
3cos
2
-3sin
+
2
2
cos 2
2
- 4sin
= 2- c2ossin22 2
2 3
The slope of the line tangent to the graph of r= 3 + 2cos
at
=
2
is
2 3
.
(c)
dr dt
= ddr
d dt
= -2 sin
d dt
d dt
= dr dt
-2
1 sin
( ) d
dt
=
3
= 3 -2si1n 3
= 3 = - 3 radians per second - 3
3 :
1 : 1 :
d=r dt dd= t
dr d
d dt
dr dt
1 -2 sin
1 : answer with units
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Visit the College Board on the Web: .
5 5 5 ?5 5 SJ 5 5 5 5
NO CALCULATOR ALLOWED
6A
y
5.
GThe graphs of the polar curves r = 4 and r = 3 + 2 cos 0 are shown in the figure above. The curves intersect
at
0
=
-n
and
0
=
5n -.
3
3
(a) GLet R be the shaded region that is inside the graph of r = 4 and also outside the graph of r = 3 + 2 cos B, as shown in the figure above. Write an expression involving an integral for the area of R.
tM / - R
IT/3
'-1
(
t ao
U-
Le)) Jd
(9-
n,3
5u - 5tr/,
?
() ?a. ................
................
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