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Unformatted text preview: AP® Calculus AB
2006 Scoring Guidelines
Form B The College Board: Connecting Students to College Success
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2006 SCORING GUIDELINES (Form B)
Question 1 x3 x 2 x
−
− + 3cos x. Let R
4
3
2
be the shaded region in the second quadrant bounded by the graph of f,
and let S be the shaded region bounded by the graph of f and line l,
the line tangent to the graph of f at x = 0, as shown above. Let f be the function given by f ( x ) = (a) Find the area of R.
(b) Find the volume of the solid generated when R is rotated about the
horizontal line y = −2. (c) Write, but do not evaluate, an integral expression that can be used
to find the area of S.
For x < 0, f ( x ) = 0 when x = −1.37312.
Let P = −1.37312. (a) Area of R = 0 ∫ P f ( x ) dx = 2.903 (b) Volume = π ∫ 0
P 2: ( ( f ( x ) + 2)2 − 4) dx = 59.361 (c) The equation of the tangent line l is y = 3 − A ⌡0 1 : integral
1 : answer ⎧ 1 : limits and constant
⎪
4 : ⎨ 2 : integrand
⎪ 1 : answer
⎩ 1
x.
2 The graph of f and line l intersect at A = 3.38987. ⎮
Area of S = ⌠ { ⎧ 1 : tangent line
⎪
3 : ⎨ 1 : integrand
⎪ 1 : limits
⎩ ((3 − 1 x ) − f ( x )) dx
2 © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES (Form B)
Question 2 Let f be the function defined for x ≥ 0 with f ( 0 ) = 5 and f ′, the () first derivative of f, given by f ′( x ) = e( − x 4 ) sin x 2 . The graph
of y = f ′( x ) is shown above. (a) Use the graph of f ′ to determine whether the graph of f is
concave up, concave down, or neither on the interval
1.7 < x < 1.9. Explain your reasoning.
(b) On the interval 0 ≤ x ≤ 3, find the value of x at which f has
an absolute maximum. Justify your answer.
(c) Write an equation for the line tangent to the graph of f at x = 2.
(a) On the interval 1.7 < x < 1.9, f ′ is decreasing and
thus f is concave down on this interval. (b) f ′( x ) = 0 when x = 0, π , 2π , 3π , K
On [0, 3] f ′ changes from positive to negative
only at π . The absolute maximum must occur at
x = π or at an endpoint. f ( 0) = 5
f ( π ) = f (0) +
f ( 3) = f ( 0 ) + ∫0 π 2: { 1 : answer
1 : reason ⎧ 1 : identifies π and 3 as candidates
⎪
 or ⎪
indicates that the graph of f
⎪
3: ⎨
increases, decreases, then increases
⎪
⎪ 1 : justifies f ( π ) > f ( 3)
⎪
⎩ 1 : answer f ′( x ) dx = 5.67911 3 ∫ 0 f ′( x ) dx = 5.57893 This shows that f has an absolute maximum at x = π . (c) f ( 2) = f ( 0) + 2 ∫ 0 f ′( x ) dx = 5.62342 ⎧ 2 : f ( 2 ) expression
⎪
1 : integral
⎪
1 : including f ( 0 ) term
4: ⎨
⎪ 1 : f ′( 2 )
⎪
⎩ 1 : equation f ′( 2 ) = e−0.5 sin ( 4 ) = −0.45902 y − 5.623 = ( −0.459 )( x − 2 ) © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES (Form B)
Question 3 The figure above is the graph of a function of x, which models the height of a
skateboard ramp. The function meets the following requirements.
(i) At x = 0, the value of the function is 0, and the slope of the graph of
the function is 0.
(ii) At x = 4, the value of the function is 1, and the slope of the graph of
the function is 1.
(iii) Between x = 0 and x = 4, the function is increasing.
(a) Let f ( x ) = ax 2 , where a is a nonzero constant. Show that it is not possible to find a value for a so that f
meets requirement (ii) above.
x2
, where c is a nonzero constant. Find the value of c so that g meets requirement (ii)
16
above. Show the work that leads to your answer.
(c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above. (b) Let g ( x ) = cx3 − xn
, where k is a nonzero constant and n is a positive integer. Find the values of k and n so that
k
h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above. (d) Let h( x ) = (a) f ( 4 ) = 1 implies that a =
implies that a = (b) 1
and f ′( 4 ) = 2a ( 4 ) = 1
16 1
. Thus, f cannot satisfy (ii).
8 1
.
32
2 ( 4)
1
1
1
When c =
, g ′( 4 ) = 3c ( 4 )2 −
=3
(16 ) − = 1
16
32
2
32 g ( 4 ) = 64c − 1 = 1 implies that c = ⎧ 1 : a = 1 or a = 1
⎪
2: ⎨
16
8
⎪ 1 : shows a does not work
⎩ 1 : value of c () (c) (d) g ′( x ) = 32 x
1
x− =
x ( 3x − 4 )
32
8 32
4
g ′( x ) < 0 for 0 < x < , so g does not satisfy (iii).
3 ⎧ 1 : g ′( x )
2: ⎨
⎩ 1 : explanation 4n
= 1 implies that 4n = k .
k
n 4n −1
n4n −1
n
h′ ( 4 ) =
=
= = 1 gives n = 4 and k = 44 = 256.
n
k
4
4 ⎧
4n
⎪1: k =1
⎪
n 1
⎪
4 : ⎨ 1 : n4
=1
k
⎪
⎪ 1 : values for k and n
⎪ 1 : verifications
⎩ h ( 4) = x4
⇒ h ( 0 ) = 0.
256
4 x3
⇒ h′ ( 0 ) = 0 and h′ ( x ) > 0 for 0 < x < 4.
h′ ( x ) =
256 h( x) = © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES (Form B)
Question 4 The rate, in calories per minute, at which a person using an
exercise machine burns calories is modeled by the function
1
3
f. In the figure above, f ( t ) = − t 3 + t 2 + 1 for
4
2
0 ≤ t ≤ 4 and f is piecewise linear for 4 ≤ t ≤ 24.
(a) Find f ′ ( 22) . Indicate units of measure.
(b) For the time interval 0 ≤ t ≤ 24, at what time t is f
increasing at its greatest rate? Show the reasoning that
supports your answer.
(c) Find the total number of calories burned over the time
interval 6 ≤ t ≤ 18 minutes.
(d) The setting on the machine is now changed so that the person burns f ( t ) + c calories per minute. For this
setting, find c so that an average of 15 calories per minute is burned during the time interval 6 ≤ t ≤ 18.
(a) f ′( 22 ) = 15 − 3
= −3 calories/min/min
20 − 24 1 : f ′( 22 ) and units (b) f is increasing on [ 0, 4] and on [12, 16].
15 − 9
3
=
since f has
On (12, 16 ) , f ′( t ) =
16 − 12 2
constant slope on this interval.
3
On ( 0, 4 ) , f ′( t ) = − t 2 + 3t and
4
3
f ′′( t ) = − t + 3 = 0 when t = 2. This is where f ′
2
has a maximum on [ 0, 4] since f ′′ > 0 on ( 0, 2 )
and f ′′ < 0 on ( 2, 4 ) . ⎧ 1 : f ′ on ( 0, 4 )
⎪ 1 : shows f ′ has a max at t = 2 on ( 0, 4 )
⎪
4: ⎨
⎪ 1 : shows for 12 < t < 16, f ′( t ) < f ′( 2 )
⎪ 1 : answer
⎩ On [ 0, 24] , f is increasing at its greatest rate when
3
t = 2 because f ′( 2 ) = 3 > .
2
1
( 4 )( 9 + 15 ) + 2 (15 )
2
= 132 calories 2: { 1 : method
1 : answer 1 18
( f ( t ) + c ) dt = 15.
12 6
This means 132 + 12c = 15(12). So, c = 4. (c) 2: { 1 : setup
1 : value of c 18 ∫6 f ( t ) dt = 6 ( 9 ) + (d) We want ∫ OR
132
= 11 calories/min.
12
Adding c to f ( t ) will shift the average by c.
So c = 4 to get an average of 15 calories/min.
Currently, the average is © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES (Form B)
Question 5 Consider the differential equation dy
= ( y − 1) 2 cos (π x ) .
dx (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.
(Note: Use the axes provided in the exam booklet.) (b) There is a horizontal line with equation y = c that satisfies this differential equation. Find the value of c.
(c) Find the particular solution y = f ( x ) to the differential equation with the initial condition f (1) = 0.
(a) 2: (b) The line y = 1 satisfies the differential equation, so
c = 1.
(c) 1
dy = cos (π x ) dx
( y − 1)2
− ( y − 1)−1 = 1 π sin (π x ) + C 1: c =1 Note: max 3 6 [12000] if no
constant of integration
Note: 0 6 if no separation of variables π 1
1
= sin (π x ) + 1
1− y π π 1 : zero slopes
1 : all other slopes ⎧ 1 : separates variables
⎪ 2 : antiderivatives
⎪
6 : ⎨ 1 : constant of integration
⎪ 1 : uses initial condition
⎪
⎩ 1 : answer 1
1
= sin (π x ) + C
1− y π
1
1 = sin (π ) + C = C 1− y { = sin (π x ) + π y =1− π
for −∞ < x < ∞
sin (π x ) + π © 2006 The College Board. All rights reserved.
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2006 SCORING GUIDELINES (Form B)
Question 6 t
(sec)
v( t )
( ft sec) ( ft a( t )
sec2 0 15 25 30 35 50 60 –20 –30 –20 –14 –10 0 10 1 5 2 1 2 4 2 ) A car travels on a straight track. During the time interval 0 ≤ t ≤ 60 seconds, the car’s velocity v, measured in
feet per second, and acceleration a, measured in feet per second per second, are continuous functions. The table
above shows selected values of these functions.
(a) Using appropriate units, explain the meaning of
60 ∫30 v( t ) dt in terms of the car’s motion. Approximate v( t ) dt using a trapezoidal approximation with the three subintervals determined by the table. (b) Using appropriate units, explain the meaning of
of 60 ∫ 30 30 ∫0 30 ∫0 a( t ) dt in terms of the car’s motion. Find the exact value a( t ) dt. (c) For 0 < t < 60, must there be a time t when v( t ) = −5 ? Justify your answer.
(d) For 0 < t < 60, must there be a time t when a( t ) = 0 ? Justify your answer.
(a) 60 ∫ 30 v( t ) dt is the distance in feet that the car travels 2: { 1 : explanation
1 : value 2: { 1 : explanation
1 : value from t = 30 sec to t = 60 sec.
Trapezoidal approximation for
A= (b) 30 ∫0 60 ∫30 v( t ) dt : 1
1
1
(14 + 10 ) 5 + (10 )(15) + (10 )(10 ) = 185 ft
2
2
2 a( t ) dt is the car’s change in velocity in ft/sec from t = 0 sec to t = 30 sec.
30 ∫0 a( t ) dt = 30 ∫0 v′( t ) dt = v ( 30 ) − v ( 0 ) = −14 − ( −20 ) = 6 ft/sec
(c) Yes. Since v( 35 ) = −10 < −5 < 0 = v( 50 ) , the IVT
guarantees a t in ( 35, 50 ) so that v( t ) = −5. ⎧ 1 : v( 35 ) < −5 < v( 50 )
2: ⎨
⎩ 1 : Yes; refers to IVT or hypotheses (d) Yes. Since v( 0 ) = v( 25 ) , the MVT guarantees a t in
( 0, 25 ) so that a( t ) = v′( t ) = 0. ⎧ 1 : v( 0 ) = v( 25 )
2: ⎨
⎩ 1 : Yes; refers to MVT or hypotheses Units of ft in (a) and ft/sec in (b) 1 : units in (a) and (b) © 2006 The College Board. All rights reserved.
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