AP CALCULUS BC 2009 SCORING GUIDELINES (Form B) - College Board
AP? CALCULUS BC 2009 SCORING GUIDELINES (Form B)
Question 4
The graph of the polar curve r = 1 - 2cos for 0 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
dx d
and
dy d
in terms of .
(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
=
2
.
Show the computations that lead to your answer.
(a)
r(0) = -1;
r( )
=
0
when
=
3
.
Area
of
S
=
1 2
0
3(1 - 2cos )2 d
{ 1 : limits and constant
2 : 1 : integrand
(b) x = r cos and y = r sin
dr d
= 2sin
dx d
=
dr d
cos
- r sin
=
4sin cos
- sin
dy d
=
dr d
sin
+ r cos
= 2sin2
+ (1 - 2cos )cos
1 : uses x = r cos and y = r sin
4
:
1
:
dr d
2 : answer
(c)
When
=
2
,
we have
x = 0,
y
= 1.
dy dx
=
2
=
dy dx
d d
= -2
=
2
The tangent line is given by y = 1 - 2x.
1 : values for x and y
3
:
1
:
expression
for
dy dx
1 : tangent line equation
? 2009 The College Board. All rights reserved. Visit the College Board on the Web: .
? 2009 The College Board. All rights reserved. Visit the College Board on the Web: .
? 2009 The College Board. All rights reserved. Visit the College Board on the Web: .
? 2009 The College Board. All rights reserved. Visit the College Board on the Web: .
? 2009 The College Board. All rights reserved. Visit the College Board on the Web: .
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