2017 AP CALCULUS AB FREE-RESPONSE QUESTIONS · 2017 AP® CALCULUS AB FREE-RESPONSE QUESTIONS 2....

2017 AP® CALCULUS AB FREE-RESPONSE QUESTIONS

2. When a certain grocery store opens, it has 50 pounds of bananas on a display table. Customers remove bananas from the display table at a rate modeled by

where f t( ) is measured in pounds per hour and t is the number of hours after the store opened. After the store has been open for three hours, store employees add bananas to the display table at a rate modeled by

g t ( ) �3 2.4 ln( 2 t � 2t ) for 3 � �t 12,

where g t( ) is measured in pounds per hour and t is the number of hours after the store opened.

(a) How many pounds of bananas are removed from the display table during the first 2 hours the store is open?

(b) Find f � 7 ( ) � 7f ( ). Using correct units, explain the meaning of in the context of the problem.

(c) Is the number of pounds of bananas on the display table increasing or decreasing at time t 5 ? Give a reason for your answer.

(d) How many pounds of bananas are on the display table at time t 8 ?

END OF PART A OF SECTION II

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GO ON TO THE NEXT PAGE. -3-

(f tt

10 sin100

3) � (0.8 t)

⎛⎜⎜⎜⎝⎜

⎞⎟⎟⎟⎠⎟⎟

for t� �0 ,12

AP® CALCULUS AB 2017 SCORING GUIDELINES


Question 2

(a) ( )

2

020.051175f t dt =∫

20.051 pounds of bananas are removed from the display table during the first 2 hours the store is open.

{ 1 : integral2 :

1 : answer

(b) ( )7 8.120f ′ = − (or 8.119− ) After the store has been open 7 hours, the rate at which bananas are being removed from the display table is decreasing by 8.120 (or 8.119) pounds per hour per hour.

{ 1 : value2 :

1 : meaning

(c) ( ) ( )5 5 2.263103 0g f− = − < Because ( ) ( )5 5 0,g f− < the number of pounds of bananas on the display table is decreasing at time 5.t =

( ) ( ) 1 : considers 5 and 52 :

1 : answer with reasonf g

(d) ( ) ( )8 8

3 050 23.347396g t dt f t dt+ − =∫ ∫

23.347 pounds of bananas are on the display table at time 8.t =

{ 2 : integrals3 :

1 : answer

AP® CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES

Question 1


The rate at which rainwater flows into a drainpipe is modeled by the function R, where ( )2

20sin 35tR t = cubic

feet per hour, t is measured in hours, and 0 8.t≤ ≤ The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by ( ) 3 20.04 0.4 0.96D t t t t= − + + cubic feet per hour, for 0 8.t≤ ≤ There are 30 cubic feet of water in the pipe at time 0.t =

(a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 0 8 ?t≤ ≤

(b) Is the amount of water in the pipe increasing or decreasing at time 3t = hours? Give a reason for your answer.

(c) At what time t, 0 8,t≤ ≤ is the amount of water in the pipe at a minimum? Justify your answer.

(d) The pipe can hold 50 cubic feet of water before overflowing. For 8,t > water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time w when the pipe will begin to overflow.

(a) ( )

8

076.570R t dt =∫ { 1 : integrand

2 : 1 : answer

(b) ( ) ( )3 3 0.313632 0R D− = − < Since ( ) ( )3 3 ,R D< the amount of water in the pipe is decreasing at time 3t = hours.

( ) ( )3 and 3 1 : considers 2 :

1 : answer and reasonR D

(c) The amount of water in the pipe at time t, 0 8,t≤ ≤ is

( ) ( )[ ]0

0 .3tR x D x dx−+ ∫

( ) ( ) 0 0, 3.271658R t D t t− = ⇒ =

t Amount of water in the pipe 0 30

3.271658 27.964561 8 48.543686

The amount of water in the pipe is a minimum at time

3.272t = (or 3.271) hours.

( ) ( ) 1 : considers 3 : 1 : answer

1 : justification

0R t D t− =

(d) ( ) ( )[ ]0

5030wR t D t dt+ − =∫ { 1 : integral

2 : 1 : equation


Question 3

© 2008 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.

Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. (Note: The volume V of a right circular cylinder with radius r and height h is given by 2 .V r hπ= ) (a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the

radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute?

(b) A recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is ( ) 400R t t= cubic centimeters per minute, where t is the time in minutes since the device began

working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time t when the oil slick reaches its maximum volume. Justify your answer.

(c) By the time the recovery device began removing oil, 60,000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).

(a) When 100r = cm and 0.5h = cm, 32000 cm mindVdt =

and 2.5 cm min.drdt =

22dV dr dhr h rdt dt dtπ π= +

( )( )( ) ( )22000 2 100 2.5 0.5 100 dhdtπ π= +

0.038 or 0.039 cm mindhdt =

4 :

1 : 2000 and 2.5

2 : expression for

1 : answer

dV drdt dt

dVdt

= =°°®°°̄

(b) ( )2000 ,dV R tdt = − so 0dVdt = when ( ) 2000.R t =

This occurs when 25t = minutes.

Since 0dVdt > for 0 25t< < and 0dV

dt < for 25,t >

the oil slick reaches its maximum volume 25 minutes after the device begins working.

3 : ( )1 : 2000

1 : answer1 : justification

R t =°®°̄

(c) The volume of oil, in 3cm , in the slick at time 25t = minutes

is given by ( )( )25

060,000 2000 .R t dt+ −³

2 : { 1 : limits and initial condition 1 : integrand

35


Question 2

© 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).

The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 7,t≤ ≤ where t is measured in hours. In this model, rates are given as follows: (i) The rate at which water enters the tank is

( ) ( )2100 sinf t t t= gallons per hour for 0 7.t≤ ≤ (ii) The rate at which water leaves the tank is

( ) 250 for 0 32000 for 3 7

tg t

t≤ <= ® < ≤¯

gallons per hour.

The graphs of f and g, which intersect at 1.617t = and 5.076,t = are shown in the figure above. At time 0,t = the amount of water in the tank is 5000 gallons.

(a) How many gallons of water enter the tank during the time interval 0 7 ?t≤ ≤ Round your answer to the nearest gallon.

(b) For 0 7,t≤ ≤ find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.

(c) For 0 7,t≤ ≤ at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.

(a) ( )7

08264f t dt ≈³ gallons

2 : { 1 : integral1 : answer

(b) The amount of water in the tank is decreasing on the

intervals 0 1.617t≤ ≤ and 3 5.076t≤ ≤ because ( ) ( )f t g t< for 0 1.617t≤ < and 3 5.076.t< <

2 : { 1 : intervals1 : reason

(c) Since ( ) ( )f t g t− changes sign from positive to negative only at 3,t = the candidates for the absolute maximum are at 0, 3,t = and 7. t (hours) gallons of water

0 5000

3 ( ) ( )3

05000 250 3 5126.591f t dt+ − =³

7 ( ) ( )7

35126.591 2000 4 4513.807f t dt+ − =³

The amount of water in the tank is greatest at 3 hours. At that time, the amount of water in the tank, rounded to the nearest gallon, is 5127 gallons.

5 :

1 : identifies 3 as a candidate 1 : integrand 1 : amount of water at 3 1 : amount of water at 7 1 : conclusion

t

tt

=°°

=®° =°¯

33


Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).

3

Question 2

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

( ) ( )42 5sin .25tR t π= +

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

( ) 15 .1 3tS t t= +

Both ( )R t and ( )S t have units of cubic yards per hour and t is measured in hours for 0 6.t≤ ≤ At time 0,t = the beach contains 2500 cubic yards of sand. (a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. (b) Write an expression for ( ) ,Y t the total number of cubic yards of sand on the beach at time t.

(c) Find the rate at which the total amount of sand on the beach is changing at time 4.t = (d) For 0 6,t≤ ≤ at what time t is the amount of sand on the beach a minimum? What is the minimum value?

Justify your answers.

(a) ( )6 30

31.815 or 31.816 ydR t dt =³

2 : { 1 : integral1 : answer with units

(b) ( ) ( ) ( )( )0

2500t

Y t S x R x dx= + −³

3 : 1 : integrand

1 : limits1 : answer

°®°̄

(c) ( ) ( ) ( )Y t S t R t′ = −

( ) ( ) ( ) 34 4 4 1.908 or 1.909 yd hrY S R′ = − = − −

1 : answer

(d) ( ) 0Y t =′ when ( ) ( ) 0.S t R t− = The only value in [ ]0, 6 to satisfy ( ) ( )S t R t= is 5.117865.a =

t ( )Y t

0 2500

a 2492.3694

6 2493.2766 The amount of sand is a minimum when 5.117t = or 5.118 hours. The minimum value is 2492.369 cubic yards.

3 : ( ) 1 : sets 0

1 : critical -value1 : answer with justification

Y tt

′ =°®°̄

31

AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B)

Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).

3

Question 2

A water tank at Camp Newton holds 1200 gallons of water at time 0.t = During the time interval 0 18t≤ ≤ hours, water is pumped into the tank at the rate

( ) ( )295 sin 6tW t t= gallons per hour.

During the same time interval, water is removed from the tank at the rate

( ) ( )2275sin 3tR t = gallons per hour.

(a) Is the amount of water in the tank increasing at time 15 ?t = Why or why not?

(b) To the nearest whole number, how many gallons of water are in the tank at time 18 ?t =

(c) At what time t, for 0 18,t≤ ≤ is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.

(d) For 18,t > no water is pumped into the tank, but water continues to be removed at the rate ( )R t until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k.

(a) No; the amount of water is not increasing at 15t = since ( ) ( )15 15 121.09 0.W R− = − <

1 : answer with reason

(b) ( ) ( )( )18

01200 1309.788W t R t dt+ − =³

1310 gallons

3 : 1 : limits1 : integrand1 : answer

°®°̄

(c) ( ) ( ) 0W t R t− = 0, 6.4948, 12.9748t =

t (hours) gallons of water 0 1200

6.495 525 12.975 1697

18 1310 The values at the endpoints and the critical points show that the absolute minimum occurs when

6.494 or 6.495. t =

3 :

1 : interior critical points 1 : amount of water is least at 6.494 or 6.4951 : analysis for absolute minimum

t

°°® =°°̄

(d) ( )18

1310k

R t dt =³

2 : 1 : limits1 : equation

®¯

30

AP® CALCULUS AB 2003 SCORING GUIDELINES (Form B)

Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com.

3

Question 2

A tank contains 125 gallons of heating oil at time 0.t = During the time interval 0 12tb b hours,

heating oil is pumped into the tank at the rate

( )( )( )

102

1 ln 1H t

t= +

+ + gallons per hour.

During the same time interval, heating oil is removed from the tank at the rate

( )2

12 sin47t

R t� ¬�= � �� ®

gallons per hour.

(a) How many gallons of heating oil are pumped into the tank during the time interval 0 12tb b hours?

(b) Is the level of heating oil in the tank rising or falling at time 6t = hours? Give a reason for your

answer.

(c) How many gallons of heating oil are in the tank at time 12t = hours?

(d) At what time t, for 0 12,tb b is the volume of heating oil in the tank the least? Show the analysis

that leads to your conclusion.

(a) 12

0( )H t dt¨ = 70.570 or 70.571 2 :

1 : integral

1 : answer

£¦¦¤¦¦¥

(b) (6) (6) 2.924,H R� = �

so the level of heating oil is falling at 6.t =

1 : answer with reason

(c) ( )12

0125 ( ) ( )H t R t dt+ �¨ = 122.025 or 122.026

3 :

1 : limits

1 : integrand

1 : answer

£¦¦¦¦¦¤¦¦¦¦¦¥

(d) The absolute minimum occurs at a critical point

or an endpoint.

( ) ( ) 0H t R t� = when 4.790t = and 11.318.t =

The volume increases until 4.790,t = then

decreases until 11.318,t = then increases, so the

absolute minimum will be at 0t = or at

11.318.t =

( )11.318

0125 ( ) ( )H t R t dt+ �¨ = 120.738

Since the volume is 125 at 0,t = the volume is

least at 11.318.t =

3 :

1 : sets ( ) ( ) 0

1 : volume is least at

11.318

1 : analysis for absolute

minimum

H t R t

t

£¦ � =¦¦¦¦¦¦¦¦ =¤¦¦¦¦¦¦¦¦¦¥

27


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

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2017 AP CALCULUS AB FREE-RESPONSE QUESTIONS · 2017 AP® CALCULUS AB FREE-RESPONSE QUESTIONS 2....

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