2002 FRQ - Scoring Guide - AB

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AP Calculus A B2002 Scoring Guidelines

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AP CALCULUS AB 2002 SCORING GUIDELINES

Copyright 2002 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Ent rance Examination Board.

2

Question 1

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

(a) Find the area of the region enclosed by the graphs of B and C between N and 1.N

(b) Find the volume of the solid generated when the region enclosed by the graphs of B and C betweenN 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 interval1

1.2

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

(a) Area = lnN A N @N = 1.222 or 1.223 2 1 : integral1 : answer

(b) Volume = 4 ln 4 N N A @N 3 = 7.515 3 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) N D 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 N a

Note: Errors in computation come off

the third point.


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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

.24 160

E 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

.38 370

L t t t

Both - J and L t are measured in people per hour and time t is measured in hours after midnight. Thesefunctions are valid for 9 23,J > > the hours during which the park is open. At time 9,J there are no people inthe 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 youranswer 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 and11 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 decreasingat the rate of approximately 380 people/hr attime 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 15.795

21 : ( ) ( ) 0

1 : answer

E t L t


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4

Question 3

An object moves along the N -axis with initial position 0 2.N The velocity of the object at time J F is given

by sin .3L J J 3

(a) What is the acceleration of the object at time 4 ?J

(b) Consider the following two statements.Statement I: For 3 4.5,J the velocity of the object is decreasing.Statement II: For 3 4.5,J the speed of the object is increasing.

Are either or both of these statements correct? For each statement provide a reason why it is correct or notcorrect.

(c) What is the total distance traveled by the object over the time interval 0 4 ?J > > (d) What is the position of the object at time 4 ?J

(a) 4(4) (4) cos3 3= L 3 3 a

=$

3 or 0.523 or 0.524

1 : answer

(b) On 3 4.5J :

( ) ( ) cos 03 3= J L J J 3 3

a

Statement I is correct since ( ) 0.= J Statement II is correct since L J and

( ) 0.= J

3

1 : I correct, with reason

1 : II correct

1 : reason for II

(c) Distance ="

( ) 2.387L J @J OR

3 3( ) cos 23N J J 3

3 3

N 9(4) 2 3.43239

2N

3

L J when J = 36(3) 2 3.90986N 3

#! " !N N N N 3

= 2.387

limits of 0 and 4 on an integral

of ( ) or ( )1 : or

uses (0) and (4) to compute

distance3

1 : handles change of direction at

student s turning point

1

L J L J

N N

: answer

0/1 if incorrect turning point or

no turning point

(d)"

(4) (0) ( ) 3.432N N L J @J OR

3 3( ) cos 23N J J 3

3 3

9(4) 2 3.4322

N 3

21 : integral

1 : answer

OR

2 31 : ( ) cos 3

1 : answer

0/1 if no constant of integration

N J J + 3

3


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5

Question 4

The 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 concavedown? 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 B a

!C B aa 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 of C 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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6

Question 5

A container has the shape of an open right circular cone, as shown in the figureabove. The height of the container is 10 cm and the diameter of the opening is 10cm. Water in the container is evaporating so that its depth D is changing at the

constant rate of !

cm/hr.

(The volume of a cone of height D and radius H is given by1

.3

8 H D 3 )

(a) Find the volume 8 of water in the container when #D cm. Indicate unitsof measure.

(b) Find the rate of change of the volume of water in the container, with respect to time, when #D cm. Indicateunits of measure.

(c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportionalto the exposed surface area of the water. What is the constant of proportionality?

(a) When D = 5,#

H ; !1 5 125(5) 5 cm3 2 128 3 3 1 : 8 when D = 5

(b)#H

D , so 1

2H D

!! "8 D D D 3 3 ; "@8 @D D @J @J 3 #D

@8

@J ! ##" &3 3 !cm

hr

OR

!@8 @D @H H HD @J @J @J 3 ; @H @D @J @J 5

5,2

3

1 25 3 5 32 5

3 4 10 2 20

15 cmhr8

D H

@8

@J 3

3

11 : in (a) or (b)2

as a function of one variable

in (a) or (b)1 : OR

5

2 :

2 chain rule or product rule error

1 : evaluation at 5

H D

8

@H @J

@8

@J

D

(c) !

" "

@8 @D D D

@J @J 3 3

! ! !

=HA="

H H 3 3

The constant of proportionality is 310

.

2

1 : shows area

1 : identifies constant of proportionality

dV k

dt

units of !cm in (a) and!cm

hr in (b)1 : correct units in (a) and (b)


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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

N 1.5 1.0 0.5 0.5 1.0 1.5

( ) B N 1 4 6 % 6 4 1

B N a % 5 3 0 3 5 7

Let B be a function that is differentiable for all real numbers. The table above gives the values of B and itsderivative B a for selected points N in the closed interval 1.5 1.5.N b b The second derivative of B has theproperty that B N aa for 1.5 1.5.N b b

(a) Evaluate 1.5

03 ( ) 4 . B N @N = Show the work that leads to your answer.

(b) Write an equation of the line tangent to the graph of B at the point where 1.N Use this line toapproximate the value of (1.2). B Is this approximation greater than or less than the actual value of (1.2)? B Give a reason for your answer.

(c) Find a positive real number H having the property that there must exist a value ? with 0 0.5? and

( ) . B ? H aa Give a reason for your answer.

(d) Let C be the function given by2 7 for 0

( )2 7 for 0.

N N N C N

N N N

p

The graph of C passes through each of the points , ( )N B N given in the table above. Is it possible that B andC are the same function? Give a reason for your answer.

(a) 1.5 1.5 1.5

0 0 0! " ! " B N @N B N @N @N a a

1.5

0

3 ( ) 3 1 ( 7) 4 1.5 24" B N N

21 : antiderivative

1 : answer

(b) # "O N (1.2) 5 0.2 4 3 B x

The approximation is less than (1.2) B because thegraph of B is concave up on the interval1 1.2N .

3

1 : tangent line

1 : computes on tangent line at 1.2

1 : answer with reason

O N

(c) By the Mean Value Theorem there is a ? with0 0.5? such that

(0.5) (0) 3 0( ) 60.5 0 0.5

B B B ? H

a aaa

2

1 : reference to MVT for (or differentiability

of )

1 : value of for interval 0 0.5

B

B

H N

a a b b

(d)

lim ( ) lim 4 1 1N N

C N N

l l

a

lim ( ) lim 4 1 1N N

C N N l l

a

Thus C a is not continuous at N = 0, but B a iscontinuous at N = 0, so B C L .

OR "C N aa for all N L , but it was shown in part

(c) that $ B ? aa for some ? L , so B C L .

2

1 : answers no with reference to

or

1 : correct reason

C C

a aa

2002 FRQ - Scoring Guide - AB

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