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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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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