2002 FRQ - Scoring Guide

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    AP

    Calculus BC

    2002 Scoring Guidelines

    These materials were produced by Educational Testing Service (ETS), which develops and administers the examinations of the Advanced Placement

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    AP

    CALCULUS BC 2002 SCORING GUIDELINES

    Copyright 2002 by College Entrance Examination Board. All rights reserved.

    Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

    2

    Question 1

    Let B and C be the functions given by NB N A and ( ) ln .C N N

    (a) Find the area of the region enclosed by the graphs ofB and C between

    N and 1.N

    (b) Find the volume of the solid generated when the region enclosed by the graphs ofB and C between

    N and N is revolved about the line 4.O

    (c) Let D be the function given by ( ) ( ) ( ).D N B N C N Find the absolute minimum value of D N on the

    closed interval1

    1,2

    N> > and find the absolute maximum value of D N on the closed interval

    11.

    2N> > Show the analysis that leads to your answers.

    (a) Area =

    lnN

    A N @N = 1.222 or 1.223 21 : integral

    1 : answer

    (b) Volume =

    4 ln 4N

    N A @N 3

    = 7.5153 or 23.609

    4

    1 : limits and constant

    2 : integrand

    1 each error

    Note: 0/2 if not of the form

    ( ) ( )

    1 : answer

    >

    =

    k R x r x dx

    (c)

    ND N B N C N AN

    a a a

    0.567143N

    Absolute minimum value and absolute

    maximum value occur at the critical point or

    at the endpoints.

    (0.567143) 2.330D

    (0.5) 2.3418D

    (1) 2.718D

    The absolute minimum is 2.330.

    The absolute maximum is 2.718.

    3

    1 : considers ( ) 0

    1 : identifies critical point

    and endpoints as candidates

    1 : answers

    D Na

    Note: Errors in computation come off

    the third point.

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    AP

    CALCULUS BC 2002 SCORING GUIDELINES

    Copyright 2002 by College Entrance Examination Board. All rights reserved.

    Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

    3

    Question 2

    The rate at which people enter an amusement park on a given day is modeled by the function E defined by

    15600.

    24 160E t

    t t

    The rate at which people leave the same amusement park on the same day is modeled by the function L defined by

    9890.

    38 370L t

    t t

    Both - J and L t are measured in people per hour and time t is measured in hours after midnight. These

    functions are valid for 9 23,J> > the hours during which the park is open. At time 9,J there are no people in

    the park.

    (a) How many people have entered the park by 5:00 P.M. ( %J )? Round answer to the nearest whole number.

    (b) The price of admission to the park is $15 until 5:00 P.M. ( %J ). After 5:00 P.M., the price of admission to

    the park is $11. How many dollars are collected from admissions to the park on the given day? Round your

    answer to the nearest whole number.

    (c) Let '

    J

    H t E x L x dx for 9 23.J> > The value of %0 to the nearest whole number is 3725.Find the value of %0= and explain the meaning of %0 and %0= in the context of the park.

    (d) At what time t, for 9 23,J> > does the model predict that the number of people in the park is a maximum?

    (a)%

    '

    ( ) 6004.270- J @ J 6004 people entered the park by 5 pm. 3

    1 : limits

    1 : integrand

    1 : answer

    (b)% !

    ' %

    15 ( ) 11 ( ) 104048.165- J @J - J @J

    The amount collected was $104,048.

    or!

    %

    ( ) 1271.283- J @ J 1271 people entered the park between 5 pm and

    11 pm, so the amount collected was

    $15 (6004) $11 (1271) $104, 041.

    1 : setup

    (c) (17) (17) (17) 380.281H E La

    There were 3725 people in the park at t =17.

    The number of people in the park was decreasing

    at the rate of approximately 380 people/hr at

    time t= 17.

    1 : value of (17)

    2 : meanings

    1 : meaning of (17)3

    1 : meaning of (17)

    1 if no reference to 17

    0

    0

    0

    J

    a

    a

    (d) H t E t L t a

    t= 15.794 or 15.7952

    1 : ( ) ( ) 0

    1 : answer

    E t L t

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    AP

    CALCULUS BC 2002 SCORING GUIDELINES

    Copyright 2002 by College Entrance Examination Board. All rights reserved.

    Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

    4

    Question 3The figure above shows the path traveled by a roller coaster

    car over the time interval &J> > seconds. The position

    of the car at time J seconds can be modeled parametrically

    by 10 4 sin , 20 1 cos ,N J J J O J J J

    where N and O are measured in meters. The derivatives of these functions are given by

    10 4 cos , 20 sin cos 1.N J J O J J J J a a

    (a) Find the slope of the path at time 2.J Show the computations that lead to your answer.(b) Find the acceleration vector of the car at the time when the cars horizontal position is N= 140.

    (c) Find the time J at which the car is at its maximum height, and find the speed, in m/sec, of the car at

    this time.

    (d) For 0 18,J there are two times at which the car is at ground level 0 .O Find these two times

    and write an expression that gives the average speed, in m/sec, of the car between these two times. Do

    not evaluate the expression.

    (a) Slope =

    (2) 18 sin 2 cos 2 1

    10 4 cos 2(2)J

    O@O

    @N N

    a

    a

    = 1.793 or 1.794

    1 : answer using@O @O @N

    @J @J @N

    (b)

    ( ) 10 4s in 140; 13.647083N J J J J

    3.529,N J O J aa aa 1.225 or 1.226

    Acceleration vector is 3.529,1.225

    or 3.529,1.226

    2

    1 : identifies acceleration vector

    as derivative of velocity vector

    1 : computes acceleration vector

    when 140N

    (c) ( ) 20 sin cos 1 0O J J J J a

    J = 3.023 or 3.024 at maximum height

    Speed =

    N J O J N J a a a

    = 6.027 or 6.028

    3

    1 : sets ( ) 0

    1 : selects first 0

    1 : speed

    O J

    J

    a

    (d) O J when 2J 3 and 4J 3

    Average speed = "

    N J O J @J

    3

    33

    = =

    "

    110 4 cos 20 sin cos 1

    2 J J J J @J

    3

    33

    3

    1 : 2 , 4

    1 : limits and constant

    1 : integrand

    J J3 3

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    AP

    CALCULUS BC 2002 SCORING GUIDELINES

    Copyright 2002 by College Entrance Examination Board. All rights reserved.

    Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

    5

    Question 4The graph of the function B shown above consists of two line segments. Let C be the

    function given by

    ( ) .N

    C N B J @J (a) Find 1 ,C 1 ,C= and 1 .C==

    (b) For what values of N in the open interval 2, 2 is C increasing? Explain yourreasoning.

    (c) For what values of N in the open interval 2, 2 is the graph of C concave

    down? Explain your reasoning.

    (d) On the axes provided, sketch the graph of C on the closed interval 2,2 .

    (a)

    !

    C B J @J B J @J

    C Ba

    !C Baa a

    3

    1 : ( 1)

    1 : ( 1)1 : ( 1)

    C

    C

    C

    a aa

    (b) C is increasing on N because

    C N B N a on this interval.2

    1 : interval

    1 : reason

    (c) The graph ofC is concave down on N

    because C N B N aa a

    on this interval.or

    because C N B N a is decreasing on this

    interval.

    21 : interval

    1 : reason

    (d)

    2

    1 : ( 2) (0) (2) 0

    1 : appropriate increasing/decreasing

    and concavity behavior1 vertical asymptote

    C C C

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    AP

    CALCULUS BC 2002 SCORING GUIDELINES

    Copyright 2002 by College Entrance Examination Board. All rights reserved.

    Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

    6

    Question 5

    Consider the differential equation 2 4 .@O

    O N@N

    (a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through

    the point (0,1) and sketch the solution curve that passes through the point (0, 1) .

    (b) Let B be the function that satisfies the given differential equation with the initial condition 0 1.B Use

    Eulers method, starting at N with a step size of 0.1, to approximate 0.2 .B Show the work that leads

    to your answer.

    (c) Find the value of > for which O N > is a solution to the given differential equation. Justify your

    answer.

    (d) Let C be the function that satisfies the given differential equation with the initial condition 0 0.C Does

    the graph of C have a local extremum at the point (0, 0)? If so, is the point a local maximum or a local

    minimum? Justify your answer.

    (a)

    2

    1 : solution curve through (0,1)

    1 : solution curve through (0, 1)

    Curves must go through the indicated

    points, follow the given slope lines, and

    extend to the boundary of the slope field.

    (b) (0.1) (0) (0)(0.1)B B B=N

    = 1 2 0 (0.1) 1.2

    (0.2) (0.1) (0.1)(0.1)

    1.2 2.4 0.4 (0.1) 1.4

    B B Bax

    x

    2

    1 : Eulers method equations or

    equivalent table applied to (at least)

    two iterations

    1 : Euler approximation to (0.2)

    (not eligible without first point)

    B

    (c) Substitute O N > in the DE:

    " N > N > , so > = 1

    ORGuess 1,> O N

    Verify: " " " @O

    O N N N @N

    .

    2 1 : uses 2 2 in DE

    1 : 1

    @N >

    @N

    >

    (d) Chas local maximum at (0, 0).

    (0,0)

    (0) 2(0) 4(0) 0,@O

    C@N

    a and

    ( ) 2 4,

    @ O @O C N

    @N@Naa so

    (0) 2 (0) 4 4 0.C Caa a

    3

    1 : (0) 0

    1 : shows (0) 4

    1 : conclusion

    C

    C

    a aa

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    AP

    CALCULUS BC 2002 SCORING GUIDELINES

    Copyright 2002 by College Entrance Examination Board. All rights reserved.

    Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

    7

    Question 6

    The Maclaurin series for the function B is given by

    "

    !

    $ " &

    ! "

    n n

    n

    xx x x x f x x

    n n

    @

    L L

    on its interval of convergence.(a) Find the interval of convergence of the Maclaurin series for B. Justify your answer.

    (b) Find the first four terms and the general term for the Maclaurin series for ( ).B N=

    (c) Use the Maclaurin series you found in part (b) to find the value of 1 .3

    B=

    (a)

    2( 1)2lim lim 2 2( 2)2

    1

    n

    nn n

    xnn x xnx

    n

    l l@ @

    N for1 1

    2 2x

    At

    N , the series is

    n

    n

    @

    which diverges since

    this is the harmonic series.

    At

    N , the series is

    n

    nn

    @

    which

    converges by the Alternating Series Test.

    Hence, the interval of convergence is

    1 1

    2 2x b

    .

    1 : sets up ratio

    1 : computes limit of ratio

    1 : identifies interior of interval

    of convergence

    2 : analysis/conclusion at endpoints5

    1 : right endpoint

    1 : left endpoint

    1 if endpoints n 1

    ot2

    1 if multiple intervals

    N

    o

    (b) ! " & $ nB N N N N Na K K2

    1 : first 4 terms

    1 : general term

    (c) The series in (b) is a geometric series.

    " & ! ! ! !

    " & $

    ! ' % !

    $

    #

    !

    n

    n

    Ba

    K K

    K K

    OR

    2( )

    1 2f x

    xa

    for

    1 1

    2 2x . Therefore,

    $! #

    !

    Ba

    2

    11 : substitutes into infinite

    3series from (b) or expresses series

    from (b) in closed form

    1 : answer for student's series

    N