2003 AP Calculus BC Scoring Guidelines - College Board
[Pages:7]AP? Calculus BC 2003 Scoring Guidelines
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AP? CALCULUS BC 2003 SCORING GUIDELINES
Question 1
Let R be the shaded region bounded by the graphs of y = x and y = e3x and the vertical line x = 1, as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal
line y = 1. (c) The region R is the base of a solid. For this solid, each cross section
perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.
Point of intersection e3x = x at (T, S) = (0.238734, 0.488604)
? (a) Area = 1( x e3x )dx T = 0.442 or 0.443
1: Correct limits in an integral in (a), (b), or (c)
2 :
???????
1 1
: :
integrand answer
( ) ? (b) Volume = Q 1 (1 e3x )2 (1 x )2 dx T = 0.453 Q or 1.423 or 1.424
3
:
???????????????????????
2 1
: :
integrand < 1 > reversal < 1 > error with constant < 1 > omits 1 in one radius < 2 > other errors answer
(c) Length = x e3x
Height = 5( x e3x )
? Volume = 1 5( x e3x )2 dx = 1.554 T
3 :
???????????????????
2 : integrand < 1 > incorrect but has x e3x as a factor
1 : answer
Copyright ? 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral..
2
AP? CALCULUS BC 2003 SCORING GUIDELINES
Question 2
A particle starts at point A on the positive x-axis at time t = 0 and travels
along the curve from A to B to C to D, as shown above. The coordinates of
( ) the particle's position x (t ), y (t ) are differentiable functions of t, where
x a(t )
=
dx dt
=
9cos
Qt 6
??
sin
Q
t+ 2
1
??
and y a(t ) = dy dt
is not explicitly given.
At time t = 9, the particle reaches its final position at point D on the positive x-axis.
(a) At point C, is dy positive? At point C, is dx positive? Give a reason for each answer.
dt
dt
(b) The slope of the curve is undefined at point B. At what time t is the particle at point B ?
( ) (c)
The line tangent to the curve at the point
x (8), y (8)
has
equation
y
=
5 9
x
2.
Find
the
velocity vector and the speed of the particle at this point.
(d) How far apart are points A and D, the initial and final positions, respectively, of the particle?
(a)
At
point
C,
dy dt
is not positive because y(t) is
decreasing along the arc BD as t increases.
At
point
C,
dx dt
is not positive because x(t) is
decreasing along the arc BD as t increases.
2 :
???????????
1 : 1 :
dy dt dx dt
not positive with reason not positive with reason
( ) (b)
dx dt
= 0;
cos
Qt 6
= 0 or sin Q
t+ 2
1 ??
=
0
Qt 6
=
Q 2
or
Q
t+ 2
1
=
Q
;
t
=
3
for both.
Particle is at point B at t = 3.
2
:
?????????
1 1
: :
sets t=
dx dt 3
=
0
( ) ( ) (c)
x a(8) = 9 cos
4Q 3
sin
3Q 2
=
9 2
y a(8) x a(8)
=
dy dx
=
5 9
y a(8)
=
5 9
x a(8)
=
5 2
The velocity vector is < 4.5,2.5 >.
Speed = 4.52 + 2.52 = 5.147 or 5.148
? (d) x(9) x(0) = 9 x a(t)dt 0 = 39.255 The initial and final positions are 39.255 apart.
3
:
????????????
1 1 1
: : :
x a(8) y a(8) speed
2
:
???????
1 1
: :
integral answer
Copyright ? 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral..
3
AP? CALCULUS BC 2003 SCORING GUIDELINES
Question 3
The figure above shows the graphs of the line
x
=
5y 3
and the curve
C
given by
x = 1 + y2 . 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 for curve C at point P. dy
(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 2 y2 = 1. Show that x 2 y2 = 1 can be written as the polar
equation
r2
=
cos2 R
1
sin2 R .
(d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar angle R that represents the area of S.
(a) At P, 5 y = 1 + y2, so y = 3 .
3
4
Since
x
=
5 3
y,
x
=
5 4
.
2
:
?????????
1 1
: :
coordinates
dx dy
at P
of
P
dx = dy
y 1 + y2
=
y. x
At P,
dx dy
=
34 54
=
3 5
.
( ) ? (b) Area =
34 0
1 + y2
5 3
y
dy
= 0.346 or 0.347
(c) x = r cos R ; y = r sin R
x 2 y2 = 1 ? r 2 cos2 R r2 sin2 R = 1
r2
=
cos2 R
1
sin2
R
3
:
????????????
1 1 1
: : :
limits integrand answer
2
:
?????????????
1 1
: substitutes x = r cos R and y = r sin R into x 2 y2 = 1 : isolates r 2
(d) Let C be the angle that segment OP makes with
the
x-axis.
Then
tan C
=
y x
=
34 54
=
3 5
.
? Area =
tan1( 35)
0
1 2
r
2
dR
? =
1 2
tan1 ( 3 5 )
0
cos2
R
1 sin2
R dR
2
:
???????
1 1
: :
limits integrand
and
constant
Copyright ? 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral..
4
AP? CALCULUS BC 2003 SCORING GUIDELINES
Question 4
Let f be a function defined on the closed interval 3 b x b 4 with f ( 0) = 3. The graph of f a, the derivative of f, consists of one line segment and a semicircle, as shown above. (a) On what intervals, if any, is f increasing? Justify your answer. (b) Find the x-coordinate of each point of inflection of the graph of f
on the open interval 3 < x < 4. Justify your answer. (c) Find an equation for the line tangent to the graph of f at the
point (0, 3).
(d) Find f (3) and f ( 4 ). Show the work that leads to your answers.
(a) The function f is increasing on [3, 2] since f a > 0 for 3 b x < 2 .
2
:
???????
1 1
: :
interval reason
(b) x = 0 and x = 2 f a changes from decreasing to increasing at x = 0 and from increasing to decreasing at x =2
(c) f a(0) = 2 Tangent line is y = 2x + 3.
2
:
???????
1 1
: :
x = 0 and x justification
=
2
only
1 : equation
? (d) f (0) f (3) = 0 f a(t)dt 3
=
1 2
(1)(1)
1 2
(2)(2)
=
3 2
f (3)
=
f (0)
+
3 2
=
9 2
? f (4) f (0) = 4 f a(t)dt 0
( ) =
8
1 2
(2)2
Q
= 8 + 2Q
f (4) = f (0) 8 + 2Q = 5 + 2Q
(( ) ) 4
:
??????????????????????????????????????????????????
1 1 1 1
:
?
1 2
2
(difference of areas
of triangles)
: answer for f (3) using FTC
:
?
8
1 2
(2)2
Q
(area of rectangle
area of semicircle)
: answer for f (4) using FTC
Copyright ? 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral..
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