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AP Calculus AB Scoring Guide

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The penguin population on an island is modeled by a differentiable function P of time t, where P(t) is thenumber of penguins and t is measured in years for 0 ≤ t ≤ 40. There are 100,000 penguins on the islandat time t = 0. The birth rate for the penguins on the island is modeled by

B(t) = 1000e0.06t penguins per year

and the death rate for the penguins on the island is modeled by

D(t)=250e0.1t penguins per year.

1. What is the rate of change for the penguin population on the island at time t = 0?

Please respond on separate paper, following directions from your teacher.

Part A

The response can earn up to 1 point:

1 point: For the correct answer

P'(0) = B(0)-D(0) = 1000 - 250 = 750 penguins per year

0 1

The response earns the following point:


P'(0) = B(0)-D(0) = 1000 - 250 = 750 penguins per year

The following are related to this scenario:

A store is having a 12-hour sale. The total number of shoppers who have entered the store t hours afterthe sale begins is modeled by the function S defined byS(t)=0.5t4−16t3+144t2 for 0 ≤ t ≤ 12 . At time t=0 ,when the sale begins, there are no shoppers in the store.




2. At what rate are shoppers entering the store 3 hours after the start of the sale?


Part A

The response can earn up to 1 point:


S ′ ( 3 )=486 shoppers/hour

0 1

The response earns the following point:


S ′ ( 3 )=486 shoppers/hour

A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in aspecial box between noon (t=0) and 8 P.M. (t=8) . The number of entries in the box t hours after noon ismodeled by a differentiable function E for 0 ≤ t ≤ 8 Values of E(t) , in hundreds of entries, at various timest are shown in the table above.

3. According to the model from part (c), at what time were the entries being processed mostquickly? Justify your answer.





Part D

1 point is earned for considering P'(t)=0

1 point is earned for identifying candidates

1 point is earned for answering with justification

P'(t)=0 when t=9.183503 and t=10.816497

Entries are being processed most quickly at time t=12

0 1 2 3

The student response earns all of the following points:

1 point is earned for considering P'(t)=0

1 point is earned for identifying candidates

1 point is earned for answering with justification

P'(t)=0 when t=9.183503 and t=10.816497




Entries are being processed most quickly at time t=12

For 0 ≤ t ≤ 31 , the rate of change of the number of mosquitoes on Tropical Island at time t days ismodeled by R(t)=5tcos(t5) mosquitoes per day. There are 1000 mosquitoes on Tropical Island at time t=0.

4. At time t=6, is the number of mosquitoes increasing at an increasing rate, or is the number ofmosquitoes increasing at a decreasing rate? Give a reason for your answer.


Part B

1 point is earned for considering R'(6)

1 point is earned for the answer with reason

R'(6)=-1.913 Since R'(6)<0, the number of mosquitoes is increasing at a decreasing rate at t=6.

0 1 2


1 point is earned for considering R'(6)

1 point is earned for the answer with reason




R'(6)=-1.913 Since R'(6)<0, the number of mosquitoes is increasing at a decreasing rate at t=6.

The rate at which people enter an auditorium for a rock concert is modeled by the function R given by R(t)= 1380t2 - 675t3 for 0≤t≤2 hours; R(t) is measured in people per hour. No one is in the auditorium at time t= 0, when the doors open. The doors close and the concert begins at time t = 2.

5. Find the time when the rate at which people enter the auditorium is a maximum. Justify youranswer.


Part B

1 point is earned if considers R'(t)=0

1 point is earned for interior critical point

1 point is earned for the answer plus justification

R′(t) = 0 when t = 0 and t = 1.36296 The maximum rate may occur at 0, a = 1.36296, or 2.

R(0)=0

R(a) = 854.527

R(2)=120

The maximum rate occurs when t=1.362 or t=1.363.

0 1 2 3

The student response earns three of the following points:




1 point is earned if considers R'(t)=0

1 point is earned for interior critical point

1 point is earned for the answer plus justification

R′(t) = 0 when t = 0 and t = 1.36296 The maximum rate may occur at 0, a = 1.36296, or 2.

R(0)=0

R(a) = 854.527

R(2)=120

The maximum rate occurs when t=1.362 or t=1.363.

6. A GRAPHING CALCULATOR IS REQUIRED FOR THIS QUESTION.

You are permitted to use your calculator to solve an equation, find the derivative of a functionat a point, or calculate the value of a definite integral. However, you must clearly indicate thesetup of your question, namely the equation, function, or integral you are using. If you useother built-in features or programs, you must show the mathematical steps necessary toproduce your results. Your work must be expressed in standard mathematical notation ratherthan calculator syntax.

Show all of your work, even though the question may not explicitly remind you to do so. Clearlylabel any functions, graphs, tables, or other objects that you use. Justifications require that yougive mathematical reasons, and that you verify the needed conditions under which relevanttheorems, properties, definitions, or tests are applied. Your work will be scored on thecorrectness and completeness of your methods as well as your answers. Answers withoutsupporting work will usually not receive credit.

Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If youranswer is given as a decimal approximation, it should be correct to three places after thedecimal point.

Unless otherwise specified, the domain of a function is assumed to be the set of all realnumbers for which is a real number.




Fish enter a lake at a rate modeled by the function given by . Fish

leave the lake at a rate modeled by the function given by . Both and are measured in fish per hour, and is measured in hours since midnight ( ).

a) How many fish enter the lake over the -hour period from midnight ( ) to A.M. () ? Give your answer to the nearest whole number.


b) What is the average number of fish that leave the lake per hour over the -hour period frommidnight ( ) to A.M. ( ) ?


c) At what time , for , is the greatest number of fish in the lake? Justify your answer.


d) Is the rate of change in the number of fish in the lake increasing or decreasing at A.M. () ? Explain your reasoning.


Part A

Select a point value to view scoring criteria, solutions, and/or examples to score the response.

The first point requires a correct integrand and limits.

The second point requires the first point to be earned. The second point is earned with an answer of




or any noninteger answer that is accurate to the number of places presented after the decimal point(up to the first 3). An answer between and inclusive, might indicate that the response usesdegree mode rather than radian mode. In this case, the second point is not earned,but the remainingpoints for the question may be earned for consistent answers using degree mode.

0 1 2

The student response accurately includes both of the criteria below.

integral

answer

Solution:

To the nearest whole number, fish enter the lake from midnight to A.M.

Part B

Select a point value to view scoring criteria, solutions, and/or examples and to score the response.

The first point requires a correct integrand and limits. The is considered for the second point. If thisintegral is computed in Part A of the question, that number can be used in place of the integral to earn thefirst point.

0 1 2


integral

answer




Solution:

The average number of fish that leave the lake per hour from midnight to A.M. is fish per hour.

Part C


The first point is earned through analytical, graphical, or verbal expressions to show where where changes sign, or where and intersect.

The second point is earned with the correct value of

The third point requires a global argument for the sign of rather than a local argument.

Another approach uses the Candidates Test. Let be the change in the number of fish in the lakefrom midnight to hours after midnight.

Therefore the greatest number of fish in the lake is at time

0 1 2 3

The student response accurately includes all three of the criteria below.




sets

answer

justification

Solution:

The rate of change in the number of fish in the lake at time is given by

for and for Therefore the greatestnumber of fish in the lake is at time

Part D


The first point requires an attempt to use and Incorrect symbolic differentiation for bothevaluated at earns the point. If values are reported without symbolic work, at least one must be correctto earn the point. These values need not be reported to three places after the decimal point.

The second point requires a direct connection between and The point is not earned ifreported values are incorrect. The numerical value of is not required, but the responsemust indicate that this value is negative.

0 1 2


considers and

answer with explanation

Solution:




Because the rate of change in the number of fish is decreasing at time

7. NO CALCULATOR IS ALLOWED FOR THIS QUESTION.




A sculptor uses a constant volume of modeling clay to form a cylinder with a large height and arelatively small radius. The clay is molded in such a way that the height of the clay increases asthe radius decreases, but it retains its cylindrical shape. At time , the height of the clay is 8inches, the radius of the clay is 3 inches, and the radius of the clay is decreasing at a rate of inch per minute.

(a) At time , at what rate is the area of the circular cross section of the clay decreasingwith respect to time? Show the computations that lead to your answer. Indicate units ofmeasure.


(b) At time , at what rate is the height of the clay increasing with respect to time? Showthe computations that lead to your answer. Indicate units of measure. (The volume of acylinder with radius and height is given by .)





(c) Write an expression for the rate of change of the radius of the clay with respect to theheight of the clay in terms of height and radius .


Part A

The first point is earned with

The second point requires substitution of values without a simplified answer.


0 1 2 3


chain rule

rate

units

Solution:

At time and




At time the area of the circular surface of the dough is decreasing at a rate of square inches perminute.

Part B

The first and second points require evidence of product rule and chain rule and no errors. For the first 2points, at most 1 out of 2 points is earned for partial communication of product rule and chain rule with amaximum of one computational error.

The third point requires substitution of values without a simplified answer.


0 1 2 3 4

The student response accurately includes all four of the criteria below.

product rule

chain rule

rate

units

Solution:

Because is constant,

At time and




At time the height of the clay is increasing at a rate of inches per minute.

Part C

The first point requires a correct expression that demonstrates chain rule with in terms of and OR in terms of and


0 1 2


chain rule

expression

Solution:


Therefore,

—OR—

Because is constant, Therefore,




8. NO CALCULATOR IS ALLOWED FOR THIS QUESTION.




A constant volume of pizza dough is formed into a cylinder with a relatively small height andlarge radius. The dough is spun and tossed into the air in such a way that the height of thedough decreases as the radius increases, but it retains its cylindrical shape. At time , theheight of the dough is inch, the radius of the dough is inches, and the radius of the doughis increasing at a rate of inches per minute.

(a) At time , at what rate is the area of the circular surface of the dough increasing withrespect to time? Show the computations that lead to your answer. Indicate units of measure.


(b) At time , at what rate is the height of the dough decreasing with respect to time?Show the computations that lead to your answer. Indicate units of measure. (The volume ofa cylinder with radius and height is given by .)


(c) Write an expression for the rate of change of the height of the dough with respect to the




radius of the dough in terms of height and radius .


Part A

The first point is earned with

The second point requires substitution of values without a simplified answer.


0 1 2 3


chain rule

rate

units

Solution:

At time and

At time the area of the circular surface of the dough is increasing at a rate of square inches perminute.

Part B




The first and second points require evidence of product rule and chain rule and no errors. For the first 2points, at most 1 out of 2 points is earned for partial communication of product rule and chain rule with amaximum of one computational error.

The third point requires substitution of values without a simplified answer.


0 1 2 3 4

The student response accurately includes all four of the criteria below.

product rule

chain rule

rate

units

Solution:


At time and

At time the height of the dough is decreasing at a rate of inch per minute.

Part C




The first point requires a correct expression that demonstrates chain rule with in terms of and OR in terms of and


0 1 2


chain rule

expression

Solution:


—OR—


9. If is the size of a population at time t, which of the following differential equations describeslinear growth in the size of the population?




A

B

C

D

E

For time t ≥ 0 hours, let r(t) = 120(1−e−10t2) represent the speed, in kilometers per hour, at which a cartravels along a straight road. The number of liters of gasoline used by the car to travel x kilometers ismodeled by g(x) = 0.05x(1-e-x/2).

10. Find the rate of change with respect to time of the number of liters of gasoline used by the carwhen t = 2 hours. Indicate units of measure.


Part B

2 points are earned for using chain rule

1 point is earned for the answer with units

dg dt = dg dx ⋅ dx dt ; dx dt =r(t) dg dt | t=2 = dg dx | x=206.370 ⋅r(2) =(0.050)(120)=6liters/hour

0 1 2 3





2 points are earned for using chain rule

1 point is earned for the answer with units

dg dt = dg dx ⋅ dx dt ; dx dt =r(t) dg dt | t=2 = dg dx | x=206.370 ⋅r(2)

=(0.050)(120)=6 liters/hour

The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by

A pumping station adds sand to the beach at a rate modeled by the function S, given by

Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for . At time t =0, the beach contains 2500 cubic yards of sand.

11. Find the rate at which the total amount of sand on the beach is changing at time t = 4.


Part C

1 point is earned for the answer

0 1


1 point is earned for the answer



Copyright © 2017. The College Board. These materials are part of a College Board program. Use or distribution of these materials online or in print beyondyour school’s participation in the program is prohibited. Page 21 of 21

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o - ApplDerivees II Q1 - Mon siterogerdurand.weebly.com/uploads/2/4/0/0/24006299/o... · in print beyond your school’s participation in the program is prohibited. Page 6 of 21 1