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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