2014 AP CALCULUS BC FREE-RESPONSE QUESTIONS...2013 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2013...

2014 AP® CALCULUS BC FREE-RESPONSE QUESTIONS

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2. The graphs of the polar curves r 3 and 3 2sin 2r q are shown in the figure above for 0 q p.

(a) Let R be the shaded region that is inside the graph of r 3 and inside the graph of 3 2sin 2r q . Find the area of R.

(b) For the curve 3 2sin 2r q , find the value of dxdq

at 6p

q .

(c) The distance between the two curves changes for 02p

q . Find the rate at which the distance between

the two curves is changing with respect to q when 3p

q .

(d) A particle is moving along the curve 3 2sin 2r q so that 3ddtq for all times 0t . Find the value

of drdt

at 6p

q .

END OF PART A OF SECTION II

AP® CALCULUS BC 2014 SCORING GUIDELINES

Question 2


The graphs of the polar curves 3r = and ( )3 2sin 2r θ= − are

shown in the figure above for 0 .θ π≤ ≤

(a) Let R be the shaded region that is inside the graph of 3r =

and inside the graph of ( )3 2sin 2 .r θ= − Find the area of R.

(b) For the curve ( )3 2sin 2 ,r θ= − find the value of dxdθ at

.6

πθ =

(c) The distance between the two curves changes for 0 .2

πθ< <

Find the rate at which the distance between the two curves is changing with respect to θ when 3

.πθ =

(d) A particle is moving along the curve ( )3 2sin 2r θ= − so that 3ddtθ = for all times 0.t ≥ Find the value

of drdt at .

6

πθ =

(a) ( )( )2

2

0

9 13 2sin 2

4 2

9.708 (or 9.707

Area

)

dππ θ θ

=

= + −∫

1 : integrand

3 : 1 : limits

1 : answer

(b) ( )( )

6

sin 2 cos

2.3

2

6

3

6dxx

d θ π

θ θ

θ == −

= −

{ 1 : expression for 2 :

1 : answer

x

(c) The distance between the two curves is

( )( ) ( )sin 23 .2 2sin 23D θ θ== − −

3

2dDd θ πθ =

= −

{ 1 : expression for distance2 :

1 : answer

(d)

( )( )6

2 3

3

6

dr dr d drd dtd d

drt

dt θ π

θθ θ

=

= ⋅ = ⋅

= − = −

{ 1 : chain rule with respect to 2 :

1 : answer

t



-3-

2. The graphs of the polar curves 3r and 4 2sinr q are shown in the figure above. The curves intersect

when 6p

q and 5 .6p

q

(a) Let S be the shaded region that is inside the graph of 3r and also inside the graph of 4 2sin .r q Find the area of S.

(b) A particle moves along the polar curve 4 2sinr q so that at time t seconds, 2.tq Find the time t in the interval 1 2t for which the x-coordinate of the particle’s position is 1.

(c) For the particle described in part (b), find the position vector in terms of t. Find the velocity vector at time 1.5.t



Question 2


The graphs of the polar curves 3r = and 4 2sinr θ= − are shown in the figure

above. The curves intersect when 6πθ = and 5 .6

πθ =

(a) Let S be the shaded region that is inside the graph of 3r = and also inside the graph of 4 2sin .r θ= − Find the area of S.

(b) A particle moves along the polar curve 4 2sinr θ= − so that at time t

seconds, 2.tθ = Find the time t in the interval 1 2t≤ ≤ for which the x-coordinate of the particle’s position is 1.−

(c) For the particle described in part (b), find the position vector in terms of t. Find the velocity vector at time 1.5.t =

(a)

( )5 6 2

61 4 2sin 24.709Area 6 2 d

π

ππ θ θ == + −∫ (or 24.708) 3 :

1 : integrand 1 : limits and constant 1 : answer

(b) ( ) ( )( ) ( )( ) ( )( )

2 2

cos 4 2sin cos

4 2sin c

1 w

os

1.428 (or 1.4hen 27)

x

x t t

t t

x

t

r

x

θ θ θ θ⇒ = −

=

=

= −

= −

3 : ( ) ( )( ) ( )

1 : or 1 : 1 or 1 1 : answer

x x tx x tθθ = − = −

(c) ( ) ( )( ) ( )( ) ( )2 2

sin 4 2sin sin

4 2sin sin

y

y t

y

t

r

t

θ θθ θ⇒ = −

= −

=

( ) ( )

( )( ) ( ) ( )( ) ( )2 2 2 2

Position vector ,

4 2 , 4sin cos sin sin2t t t t

x t y t=

= − −

( ) ( ) ( )

( )1.5 1.5 , 1.5

8.072, 1.673 or 8.072, 1.672v x y′ ′=

= − − − −

3 : { 2 : position vector 1 : velocity vector

2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)

© 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.

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CALCULUS BC SECTION II, Part B

Time—45 minutes Number of problems—3

No calculator is allowed for these problems.

4. The graph of the polar curve 1 2cosr q= - for 0 q p£ £ is shown above. Let S be the shaded region in the third quadrant bounded by the curve and the x-axis.

(a) Write an integral expression for the area of S.

(b) Write expressions for dxdq

and dydq

in terms of .q

(c) Write an equation in terms of x and y for the line tangent to the graph of the polar curve at the point

where .2p

q = Show the computations that lead to your answer.

WRITE ALL WORK IN THE EXAM BOOKLET.

AP® CALCULUS BC 2009 SCORING GUIDELINES (Form B)

Question 4

© 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.

The graph of the polar curve 1 2cosr θ= − for 0 θ π≤ ≤ is shown above. Let S be the shaded region in the third quadrant bounded by the curve and the x-axis.

(a) Write an integral expression for the area of S.

(b) Write expressions for dxdθ and dydθ in terms of .θ

(c) Write an equation in terms of x and y for the line tangent to

the graph of the polar curve at the point where .2πθ =

Show the computations that lead to your answer.

(a) ( )0 1;r = − ( ) 0r θ = when .3πθ =

Area of ( )3 2

01 1 2cos2S d

πθ θ= −∫

2 : { 1 : limits and constant 1 : integrand

(b) cosx r θ= and siny r θ=

2sindrd θθ =

cos sin 4sin cos sindx dr rd d θ θ θ θ θθ θ= − = −

( )2sin cos 2sin 1 2cos cosdy dr rd d θ θ θ θ θθ θ= + = + −

4 :

1 : uses cos and sin

1 :

2 : answer

x r y rdrd

θ θ

θ

= =⎧⎪⎨⎪⎩

(c) When ,2πθ = we have 0, 1.x y= =

2 2

2dy ddydx dx dπ πθ θ

θθ= =

= = −

The tangent line is given by 1 2 .y x= −

3 :

1 : values for and

1 : expression for

1 : tangent line equation

x ydydx

⎧⎪⎪⎨⎪⎪⎩


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-4-

3. The graphs of the polar curves 2=r and 3 2cos q= +r are shown in the figure above. The curves intersect

when 23p

q = and 4 .3p

q =

(a) Let R be the region that is inside the graph of 2=r and also inside the graph of 3 2cos ,q= +r as shaded in the figure above. Find the area of R.

(b) A particle moving with nonzero velocity along the polar curve given by 3 2cos q= +r has position

( ) ( )( ),x t y t at time t, with 0q = when 0.=t This particle moves along the curve so that .q

=dr drdt d

Find

the value of drdt

at 3p

q = and interpret your answer in terms of the motion of the particle.

(c) For the particle described in part (b), .q

=dy dydt d

Find the value of dydt

at 3p

q = and interpret your answer

in terms of the motion of the particle.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.



Question 3

© 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).

The graphs of the polar curves 2r = and 3 2cosr θ= + are shown in

the figure above. The curves intersect when 23πθ = and 4 .3

πθ =

(a) Let R be the region that is inside the graph of 2r = and also inside the graph of 3 2cos ,r θ= + as shaded in the figure above. Find the area of R.

(b) A particle moving with nonzero velocity along the polar curve given by 3 2cosr θ= + has position ( ) ( )( ),x t y t at time t, with 0θ =

when 0.t = This particle moves along the curve so that .dr drdt dθ=

Find the value of drdt at 3πθ = and interpret your answer in terms of the motion of the particle.

(c) For the particle described in part (b), .dy dydt dθ= Find the value of dydt at 3

πθ = and interpret your

answer in terms of the motion of the particle. (a)

Area ( ) ( )4 32 22 3

2 12 3 2cos3 210.370

dπ

ππ θ θ= + +

=∫

4 :

1 : area of circular sector2 : integral for section of limaçon

1 : integrand 1 : limits and constant 1 : answer

⎧⎪⎪⎨⎪⎪⎩

(b) 3 3

1.732dr drdt dθ π θ πθ= =

= = −

The particle is moving closer to the origin, since 0drdt <

and 0r > when .3πθ =

2 : 3 1 :

1 : interpretation

drdt θ π=

⎧⎪⎨⎪⎩

(c) ( )sin 3 2cos siny r θ θ θ= = +

3 30.5dy dy

dt dθ π θ πθ= == =

The particle is moving away from the x-axis, since

0dydt > and 0y > when .3

πθ =

3 : 3

1 : expression for in terms of

1 :

1: interpretation

ydydt θ π

θ

=

⎧⎪⎪⎨⎪⎪⎩


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GO ON TO THE NEXT PAGE.

3

2. The curve above is drawn in the xy-plane and is described by the equation in polar coordinates ( )sin 2r q 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 given by ( )1 2cos 2 .drd

qq= +

(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 ,3 3p p

q< < drdq

is negative. What does this fact say about r ? What does this fact say about the

curve?

(d) Find the value of q in the interval 02p

q£ £ 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.


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3

Question 2

The curve above is drawn in the xy-plane and is described by the equation in polar coordinates ( )sin 2r θ θ= + for 0 ,θ π≤ ≤ where r is measured in meters and θ is measured in radians. The derivative of r with respect to θ is

given by ( )1 2cos 2 .drd

θθ = +

(a) Find the area bounded by the curve and the x-axis. (b) Find the angle θ that corresponds to the point on the curve with

x-coordinate 2.−

(c) For 2 ,3 3π πθ< < dr

dθ is negative. What does this fact say about r ? What does this fact say about the curve?

(d) Find the value of θ in the interval 0 2πθ≤ ≤ that corresponds to the point on the curve in the first quadrant

with greatest distance from the origin. Justify your answer.

(a) Area

( )( )

20

20

121 sin 2 4.3822

r d

d

π

π

θ

θ θ θ

=

= + =

∫

∫

3 : 1 : limits and constant

1 : integrand 1 : answer

⎧⎪⎨⎪⎩

(b) ( ) ( )( ) ( )2 cos sin 2 cosr θ θ θ θ− = = + 2.786θ =

2 : { 1 : equation1 : answer

(c) Since 0drdθ < for 2 ,3 3

π πθ< < r is decreasing on this

interval. This means the curve is getting closer to the origin.

2 : { 1 : information about 1 : information about the curve

r

(d) The only value in 0, 2π⎡ ⎤

⎢ ⎥⎣ ⎦ where 0dr

dθ = is .3πθ =

θ r 0 0

3π 1.913

2π 1.571

The greatest distance occurs when .3πθ =

2 : 1 : or 1.04731 : answer with justification

πθ⎧ =⎪⎨⎪⎩


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students and parents at www.collegeboard.com/apstudents.

4

3. The figure above shows the graphs of the line x y! 53 and the curve C given by x y! "1 2 . Let S be the

shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P.

(a) Find the coordinates of point P and the value of dxdy

for curve C at point P.

(b) Set up and evaluate an integral expression with respect to y that gives the area of S.

(c) Curve C is a part of the curve x y2 2 1# ! . Show that x y2 2 1# ! can be written as the polar equation

r 22 2

1!#cos sin

.

(d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar angle that represents the area of S.


AP® CALCULUS BC

2003 SCORING GUIDELINES

Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com.

4

Question 3

The figure above shows the graphs of the line

53

x y= and the curve C given by

21 .x y= + Let S be the shaded region bounded by the two graphs and the

x-axis. The line and the curve intersect at point P.

(a) Find the coordinates of point P and the value of dx

dy for curve C at point P.

(b) Set up and evaluate an integral expression with respect to y that gives the

area of S.

(c) Curve C is a part of the curve 2 2 1.x y = Show that 2 2 1x y = can be written as the polar

equation 22 2

1.

cos sinr =

(d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar

angle that represents the area of S.

(a) At P, 251 ,

3y y= + so

3.

4y =

Since 5 5

, .3 4

x y x= =

2

.1

dx y y

dy xy= =

+ At P,

3 34 .5 54

dx

dy= =

2 :

1 : coordinates of

1 : at

P

dxP

dy

(b) Area = ( )34 2

0

51

3y y dy+

= 0.346 or 0.347

3 :

1 : limits

1 : integrand

1 : answer

(c) cosx r= ; siny r=

2 2 2 2 2 21 cos sin 1x y r r= =

22 2

1

cos sinr =

2 : 2 2

2

1 : substitutes cos and

sin into 1

1 : isolates

x r

y r x y

r

=

= =

(d) Let be the angle that segment OP makes with

the x-axis. Then 3 34tan .5 54

y

x= = =

( )

( )

1 3tan 5 2

0

1 3tan 52 20

1Area

2

1 12 cos sin

r d

d

=

=

2 : 1 : limits

1 : integrand and constant

2014 AP CALCULUS BC FREE-RESPONSE QUESTIONS...2013 AP® CALCULUS BC FREE-RESPONSE QUESTIONS © 2013...

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