AP Calculus BC - College Board
2018
AP Calculus BC
Sample Student Responses and Scoring Commentary
Inside:
Free Response Question 5 RR Scoring Guideline RR Student Samples RR Scoring Commentary
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AP? CALCULUS BC 2018 SCORING GUIDELINES
Question 5
( ) (a)= Area
1 2
5 3 3
42
- (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
cos2
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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5 5 5 ?5 5 SJ 5 5 5 5
NO CALCULATOR ALLOWED
6A
y
5. The 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 3
and
0
=
- 53n .
(a) Let 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.
R
tM / - IT/3 '-1
(
t
U-
aoLe)) Jd
(9-
n,3
5u - 5tr/,
?
() ?a. ................
................
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