2014 ap calculus bc free-response questions...2013 ap® calculus bc free-response questions © 2013...
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2014 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
© 2014 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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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
© 2014 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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
2013 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.
-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
END OF PART A OF SECTION II
AP® CALCULUS BC 2013 SCORING GUIDELINES
Question 2
© 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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
⎧⎪⎪⎨⎪⎪⎩
2007 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
© 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
-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.
END OF PART A OF SECTION II
AP® CALCULUS BC 2007 SCORING GUIDELINES
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
=
⎧⎪⎪⎨⎪⎪⎩
2005 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright © 2005 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).
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.
AP® CALCULUS BC 2005 SCORING GUIDELINES
Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).
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⎧ =⎪⎨⎪⎩
2003 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available to AP professionals at apcentral.collegeboard.com and to
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.
END OF PART A OF SECTION II
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