AP® CALCULUS AB

CALCULUS AB

2010 SCORING GUIDELINES

Question 2

(hours)

0 2 5 7 8

(

)

(hundreds of

entries)

4 13 21 23

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A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box

between noon

()

t = 0 and 8 P.M.

()

t = 8. The number of entries in the box t hours after noon is modeled by a

differentiable function E for 0 8.t≤≤ Values of E

(

)

, in hundreds of entries, at various times t are shown in

the table above.

(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being

deposited at time t = 6. Show the co

mputations that lead to your answer.

(b)

Use a trapezoidal sum with the four subintervals given by the table to approximate the value of

()

E tdt

∫

Using correct units, explain the meaning of

()

E tdt

∫

in terms of the number of entries.

P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function

P, where

()

30 298 976Pt t t t=− + − hundreds of entries per hour for

8t≤≤12.

According to the model,

how many entries had not yet been processed by midnight

(

t = 12 ?

)

(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify

your answer.

(a)

()

() ()

−

′

≈

−

hundred entries per hour

1 : answer

(b)

()

Et dt≈

∫

02 25 57 7

2· 3· 2· 1·

82 2 2

EE EE EE E+++

⎛

+++

⎜

⎝

() () () () () () () ()

⎞

⎟

⎠

10.687 or 10.868

()

E t

∫

is the average number of hundreds of entries in the box

between noon and 8

P.M.

3 :

⎧

⎪

⎨

⎪

⎩

1 : trapezoidal sum

1 : approximation

1 : meaning

(c)

()

23 23 16 7Pt dt−=−=

∫

hundred entries

2 :

{

1 : integral

1 : answer

(d)

()

0Pt

′

when t = 9.183503 and t = 10.816497.

tP(t)

9.183503 5.088662

10.816497 2.911338

12 8

Entries are being processed most quickly at time

t = 12.

3 :

⎧

⎪

⎨

⎪

⎩

1 : considers

()

0Pt

′

1 : identifies candidates

1 : answer with justification

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CALCULUS AB

2010 SCORING COMMENTARY

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Question 2

Overview

This problem involved a zoo’s contest to name a baby elephant. Students were presented with a table of values

indicating the number of entries

(

)

measured in hundreds, received in a special box and recorded at various

times t during an eight-hour period. Part (a) asked for an estimate of the rate of deposit of entries into the box at

time 6.t = Students needed to recognize this rate to be the derivative value

(

′

)

. Since t = 6 falls between the

time values specified in the table, students needed to calculate the average rate of change of E across the smallest

time subinterval from the table that brackets

t = 6. Part (b) asked for an approximation to

()

E tdt



using a

trapezoidal sum and the subintervals of [0, 8] indicated by the data in the table. Students were further asked to

interpret this expression in context, with the expectation that they would recognize that it gives the average

number of hundreds of entries in the box during the eight-hour period. In part (c) a function P was supplied that

models the rate at which entries from the box were processed, by the hundred, during a four-hour period

(

8≤≤t

)

that began after all entries had been received. This part asked for the number of entries that remained

to be processed after the four hours. Students needed to recognize that the number of entries processed is given by

()

Pt d



t, so that the number remaining to be processed, in hundreds of entries, is given by the difference

between the total number of entries in the box,

(

)

as given by the table, and the value of this integral. Part (d)

cited the model P

(

)

introduced in the previous part and asked for the time at which the entries were being

processed most quickly. Students should have recognized this as asking for the time corresponding to the

maximum value of

(

)

on the interval 8 12t≤≤ and applied a standard process for optimization on a closed

interval.

Sample: 2A

Score: 9

The student earned all 9 points.

Sample: 2B

Score: 6

The student earned 6 points: 1 point in part (a), 2 points in part (b), 2 points in part (c), and 1 point in part (d). In

part (a) the student sets up a correct difference quotient based on the values in the table and correctly evaluates for

the numerical answer. In part (b) the student sets up a correct trapezoidal sum and evaluates it based on the data in

the table to obtain a correct approximation. The student did not earn the third point in part (b) because the meaning

given does not address the time interval over which the average was computed. In part (c) the student earned both

points. The first point was earned for correctly providing the definite integral that represents the number of

hundreds of entries processed between 8

P.M. and midnight. The second point was earned for subtracting that

value from the initial condition of 23 hundred entries in the box at 8

P.M. to obtain the answer. In part (d) the

student earned the first point for setting

()

0.Pt

′

= The student does not consider the endpoints as candidates, so

no additional points were earned.

CALCULUS AB

2010 SCORING COMMENTARY

Question 2 (continued)

Sample: 2C

Score: 3

The student earned 3 points: 1 point in part (a), no points in part (b), 2 points in part (c), and no points in part (d).

In part (a) the student sets up a correct difference quotient based on the values in the table and correctly evaluates for

the numerical answer. In part (b) the student subtracts the function values at endpoints of the subintervals rather

than adding them. The student interprets the integral expression as an average rate rather than an average number.

In part (c) the student’s work is correct. “700” was accepted because of the units in this question. In part (d) the

student never considers

′

()

, so no points were earned.

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