洋桃课堂AP微积分2005真题

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AP

Calculus BC

2005 Free-Response Questions

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2005 AP

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CALCULUS BC FREE-RESPONSE QUESTIONS

CALCULUS BC

SECTION II, Part A

Time

—

45 minutes

Number of problems

—

3

A graphing calculator is required for some problems or parts of problems.

1

-

x

1.

Let

f

and

g

be the functions given by

f

(

x

)

=

4

+

sin

(

p

x

)

and

g

(

x

)

=

4

. Let

R

be the shaded region in

the first quadrant enclosed by the

y

-axis and the graphs of

f

and

g

, and let

S

be the shaded region in the first

quadrant enclosed by the graphs of

f

and

g

, as shown in the figure above.

(a)

Find the area of

R

.

(b)

Find the area of

S

.

(c)

Find the volume of the solid generated when

S

is revolved about the horizontal line

y

= -

1.

WRITE ALL WORK IN THE TEST BOOKLET.

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2

2005 AP

?

CALCULUS BC FREE-RESPONSE QUESTIONS

2. The curve above is drawn in the

xy

-plane and is described by the equation in polar coordinates

r

=

q

+

sin

(

2

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

dr

given by

d

q

=

1

+

2cos

(

2

q

)

.

(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

p

dr

p

,

is negative. What does this fact say about

r

? What does this fact say about the

<

q

<

3

3

d

q

curve?

p

(d)

Find the value of

q

in the interval 0

?

q

?

2

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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Visit (for AP professionals) and /apstudents (for AP students and parents).

GO ON TO THE NEXT PAGE.

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