Unformatted text preview: AP® Calculus AB
2006 Scoring Guidelines The College Board: Connecting Students to College Success
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AP Central is the official online home for the AP Program: apcentral.collegeboard.com. AP® CALCULUS AB
2006 SCORING GUIDELINES
Question 1
Let R be the shaded region bounded by the graph of y = ln x and the line
y = x − 2, as shown above.
(a) Find the area of R.
(b) Find the volume of the solid generated when R is rotated about the horizontal
line y = −3. (c) Write, but do not evaluate, an integral expression that can be used to find the
volume of the solid generated when R is rotated about the yaxis. ln ( x ) = x − 2 when x = 0.15859 and 3.14619.
Let S = 0.15859 and T = 3.14619 (a) Area of R = T ∫ S ( ln ( x ) − ( x − 2 ) ) dx = 1.949 (b) Volume = π ∫ T
S ⎧ 1 : integrand
⎪
3 : ⎨ 1 : limits
⎪ 1 : answer
⎩ ( ( ln ( x ) + 3)2 − ( x − 2 + 3)2 ) dx 3: { 2 : integrand
1 : limits, constant, and answer 3: { 2 : integrand
1 : limits and constant = 34.198 or 34.199 ⎮
(c) Volume = π ⌠ T −2 ⌡S − 2 ( ( y + 2) 2 () − ey 2 ) dy © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES
Question 2 At an intersection in Thomasville, Oregon, cars turn
t
cars per hour
left at the rate L( t ) = 60 t sin 2
3
over the time interval 0 ≤ t ≤ 18 hours. The graph of
y = L( t ) is shown above.
(a) To the nearest whole number, find the total
number of cars turning left at the intersection
over the time interval 0 ≤ t ≤ 18 hours.
(b) Traffic engineers will consider turn restrictions
when L( t ) ≥ 150 cars per hour. Find all values
of t for which L( t ) ≥ 150 and compute the
average value of L over this time interval.
Indicate units of measure.
(c) Traffic engineers will install a signal if there is any twohour time interval during which the product of the
total number of cars turning left and the total number of oncoming cars traveling straight through the
intersection is greater than 200,000. In every twohour time interval, 500 oncoming cars travel straight
through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to
your conclusion. () 18 (a) ∫0 (b) L ( t ) = 150 when t = 12.42831, 16.12166
Let R = 12.42831 and S = 16.12166
L ( t ) ≥ 150 for t in the interval [ R, S ]
S
1
∫ R L ( t ) dt = 199.426 cars per hour
S−R L ( t ) dt ≈ 1658 cars 2: { 1 : setup
1 : answer ⎧ 1 : t interval when L ( t ) ≥ 150
⎪
3 : ⎨ 1 : average value integral
⎪ 1 : answer with units
⎩ (c) For the product to exceed 200,000, the number of cars
turning left in a twohour interval must be greater than 400.
15 ∫13 L ( t ) dt = 431.931 > 400 ⎧ 1 : considers 400 cars
⎪ 1 : valid interval [ h, h + 2]
⎪
h+2
4: ⎨
⎪ 1 : value of ∫ h L ( t ) dt
⎪
⎩ 1 : answer and explanation OR OR
The number of cars turning left will be greater than 400
on a twohour interval if L( t ) ≥ 200 on that interval.
L( t ) ≥ 200 on any twohour subinterval of
[13.25304, 15.32386]. ⎧ 1 : considers 200 cars per hour
⎪ 1 : solves L( t ) ≥ 200
⎪
4: ⎨
⎪ 1 : discusses 2 hour interval
⎪ 1 : answer and explanation
⎩ Yes, a traffic signal is required.
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2006 SCORING GUIDELINES
Question 3 The graph of the function f shown above consists
of six line segments. Let g be the function given by
g( x) = x ∫0 f ( t ) dt. (a) Find g ( 4 ) , g ′( 4 ) , and g ′′( 4 ) .
(b) Does g have a relative minimum, a relative
maximum, or neither at x = 1 ? Justify your
answer. (c) Suppose that f is defined for all real numbers x and is periodic with a period of length 5. The graph above
shows two periods of f. Given that g ( 5 ) = 2, find g (10 ) and write an equation for the line tangent to the
graph of g at x = 108. (a) g ( 4) = 4 ∫0 f ( t ) dt = 3 ⎧ 1 : g ( 4)
⎪
3 : ⎨ 1 : g ′( 4 )
⎪ 1 : g ′′( 4 )
⎩ g ′( 4 ) = f ( 4 ) = 0
g ′′( 4 ) = f ′( 4 ) = −2 (b) g has a relative minimum at x = 1
because g ′ = f changes from negative to positive at
x = 1. (c) g ( 0 ) = 0 and the function values of g increase by 2 for
every increase of 5 in x.
g (10 ) = 2 g ( 5 ) = 4 g (108 ) = 105 ∫0 f ( t ) dt + 108 ∫105 f ( t ) dt 2: { 1 : answer
1 : reason ⎧ 1 : g (10 )
⎪
⎪
4: ⎨
⎧ 1 : g (108 )
⎪ 3 : ⎪ 1 : g ′(108 )
⎨
⎪
⎪
⎩
⎩ 1 : equation of tangent line = 21g ( 5 ) + g ( 3) = 44
g ′(108 ) = f (108 ) = f ( 3) = 2
An equation for the line tangent to the graph of g at
x = 108 is y − 44 = 2 ( x − 108 ) .
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2006 SCORING GUIDELINES
Question 4 t
(seconds) 0 10 20 30 40 50 60 70 80 v( t )
(feet per second) 5 14 22 29 35 40 44 47 49 Rocket A has positive velocity v( t ) after being launched upward from an initial height of 0 feet at time t = 0
seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 ≤ t ≤ 80 seconds, as
shown in the table above.
(a) Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate units of
measure.
(b) Using correct units, explain the meaning of 70 ∫10 v( t ) dt in terms of the rocket’s flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate 70 ∫10 v( t ) dt. 3
feet per second per second. At time
t +1
t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of
the two rockets is traveling faster at time t = 80 seconds? Explain your answer. (c) Rocket B is launched upward with an acceleration of a( t ) = (a) Average acceleration of rocket A is 1 : answer v( 80 ) − v( 0 ) 49 − 5 11
=
=
ft sec 2
80 − 0
80
20
(b) Since the velocity is positive, 70 ∫10 v( t ) dt represents the distance, in feet, traveled by rocket A from t = 10 seconds
to t = 70 seconds. ⎧ 1 : explanation
⎪
3 : ⎨ 1 : uses v( 20 ) , v( 40 ) , v( 60 )
⎪ 1 : value
⎩ A midpoint Riemann sum is
20 [ v( 20 ) + v( 40 ) + v( 60 )]
= 20 [ 22 + 35 + 44] = 2020 ft
(c) Let vB ( t ) be the velocity of rocket B at time t.
⌠ 3 dt = 6 t + 1 + C
vB ( t ) = ⎮
⌡ t +1
2 = vB ( 0 ) = 6 + C ⎧ 1: 6 t +1
⎪ 1 : constant of integration
⎪
⎪
4 : ⎨ 1 : uses initial condition
⎪ 1 : finds v ( 80 ) , compares to v( 80 ) ,
B
⎪
and draws a conclusion
⎪
⎩ vB ( t ) = 6 t + 1 − 4
vB ( 80 ) = 50 > 49 = v( 80 ) Rocket B is traveling faster at time t = 80 seconds.
1 : units in (a) and (b) Units of ft sec2 in (a) and ft in (b) © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES
Question 5 dy 1 + y
=
, where x ≠ 0.
dx
x
(a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.
(Note: Use the axes provided in the pink exam booklet.) Consider the differential equation (b) Find the particular solution y = f ( x ) to the differential equation with the initial condition f ( −1) = 1 and
state its domain.
(a) (b) 2 : sign of slope at each point and relative
steepness of slope lines in rows and
columns 1
1
dy = dx
1+ y
x ⎧ 1 : separates variables
⎧
⎪ 2 : antiderivatives
⎪
⎪
⎪
⎪ 6 : ⎨ 1 : constant of integration
⎪ 1 : uses initial condition
⎪
⎪
⎪
⎪
⎩ 1 : solves for y
7: ⎨
Note: max 3 6 [12000] if no
⎪
⎪
constant of integration
⎪
⎪ Note: 0 6 if no separation of variables
⎪
⎪ 1 : domain
⎩ ln 1 + y = ln x + K 1 + y = eln x + K
1+ y = C x
2=C
1+ y = 2 x
y = 2 x − 1 and x < 0
or
y = −2 x − 1 and x < 0 © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES
Question 6 The twicedifferentiable function f is defined for all real numbers and satisfies the following conditions:
f ( 0) = 2, f ′( 0) = − 4, and f ′′ ( 0) = 3.
(a) The function g is given by g ( x ) = eax + f ( x ) for all real numbers, where a is a constant. Find g ′ ( 0) and
g ′′ ( 0) in terms of a. Show the work that leads to your answers.
(b) The function h is given by h( x ) = cos ( kx ) f ( x ) for all real numbers, where k is a constant. Find h′ ( x ) and
write an equation for the line tangent to the graph of h at x = 0.
(a) g ′( x ) = aeax + f ′( x )
g ′( 0 ) = a − 4 ⎧ 1 : g ′( x )
⎪ 1 : g ′( 0 )
⎪
4: ⎨
⎪ 1 : g ′′( x )
⎪ 1 : g ′′( 0 )
⎩ g ′′( x ) = a 2 eax + f ′′( x )
g ′′( 0 ) = a 2 + 3 (b) h′( x ) = f ′( x ) cos ( kx ) − k sin ( kx ) f ( x )
h′( 0 ) = f ′( 0 ) cos ( 0 ) − k sin ( 0 ) f ( 0 ) = f ′( 0 ) = − 4
h( 0 ) = cos ( 0 ) f ( 0 ) = 2
The equation of the tangent line is y = − 4 x + 2. ⎧ 2 : h′( x )
⎪
⎪
⎧ 1 : h′( 0 )
5: ⎨
⎪
⎪ 3 : ⎨ 1 : h( 0 )
⎪
⎪
⎩
⎩ 1 : equation of tangent line © 2006 The College Board. All rights reserved.
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