AP CALCULUS AB/CALCULUS BC 2014 SCORING GUIDELINES

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1 AP CALCULUS AB/CALCULUS BC 14 SCORING GUIDELINES Question 1 Grass clippings are placed in a bin, where they decompose. For t 3, the amount of grass clippings remaining in the bin is modeled by At ( ) = 6.687(.931 ), where At () is measured in pounds and t is measured in days. (a) Find the average rate of change of At ( ) over the interval t 3. Indicate units of measure. (b) Find the value of A ( 15 ). Using correct units, interpret the meaning of the value in the context of the problem. (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval t 3. (d) For t > 3, Lt ( ), the linear approximation to A at t = 3, is a better model for the amount of grass clippings remaining in the bin. Use Lt ( ) to predict the time at which there will be.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer. t (a) A( 3) A( ) 3 =.197 (or.196) lbs/day 1 : answer with units (b) A ( 15 ) = (or.163) The amount of grass clippings in the bin is decreasing at a rate of.164 (or.163) lbs/day at time t = 15 days. 1 : A ( 15) : 1 : interpretation (c) 1 3 At ( ) = At ( ) dt t (or ) 3 = : A( t) dt : 3 1 : answer (d) Lt ( ) = A( 3) + A ( 3) ( t 3) A ( 3) = A( 3) =.7898 : expression for Lt ( ) 4 : 1 : Lt ( ) =.5 1 : answer Lt ( ) =.5 t= The College Board. Visit the College Board on the Web:

7 For time AP CALCULUS AB 8 SCORING GUIDELINES (Form B) 1t t hours, let rt () 1( 1 e ) Question = represent the speed, in kilometers per hour, at which a car travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is g x =.5x 1 e x. modeled by ( ) ( ) (a) How many kilometers does the car travel during the first hours? (b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when t = hours. Indicate units of measure. (c) How many liters of gasoline have been used by the car when it reaches a speed of 8 kilometers per hour? kilometers : { 1 : integral (a) r() t dt = : answer dg dg dx dx (b) = ; = rt () dt dx dt dt dg dg r( ) dt = dx t= x= 6.37 = (.5)( 1) = 6 liters hour : uses chain rule 3 : { 1 : answer with units (c) Let T be the time at which the car s speed reaches 8 kilometers per hour. Then, rt ( ) = 8 or T = hours. 4 : 1 : equation rt () = 8 : distance integral 1 : answer At time T, the car has gone T xt ( ) = r( t) dt= kilometers and has consumed g( x( T )) =.537 liters of gasoline. 8 The College Board. All rights reserved. Visit the College Board on the Web: 34

8 At an intersection in Thomasville, Oregon, cars turn t left at the rate Lt () = 6 tsin ( 3) cars per hour over the time interval t 18 hours. The graph of y = L() t is shown above. (a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval t 18 hours. (b) Traffic engineers will consider turn restrictions when Lt () 15 cars per hour. Find all values AP CALCULUS AB 6 SCORING GUIDELINES Question of t for which Lt () 15 and compute the average value of L over this time interval. Indicate units of measure. (c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than,. In every two-hour time interval, 5 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion. 18 (a) Lt () dt 1658 cars (b) Lt () = 15 when t = , Let R = and S = Lt () 15 for t in the interval [ R, S ] 1 S L () t dt = S R cars per hour R (c) For the product to exceed,, the number of cars turning left in a two-hour interval must be greater than OR Lt () dt= > 4 The number of cars turning left will be greater than 4 on a two-hour interval if Lt () on that interval. Lt () on any two-hour subinterval of [ , ]. : { 1 : setup 1 : answer 3 : 1 : t-interval when L() t 15 1 : average value integral 1 : answer with units 1 : considers 4 cars 1 : valid interval [ h, h ] + 4 : h+ 1 : value of Lt () dt h 1 : answer and explanation 4 : OR 1 : considers cars per hour 1 : solves Lt () 1 : discusses hour interval 1 : answer and explanation Yes, a traffic signal is required. 6 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 3 3

9 AP CALCULUS AB 5 SCORING GUIDELINES (Form B) Question A water tank at Camp Newton holds 1 gallons of water at time t =. During the time interval t 18 hours, water is pumped into the tank at the rate t () 95 tsin ( ) W t = gallons per hour. 6 During the same time interval, water is removed from the tank at the rate t () 75sin ( ) Rt = gallons per hour. 3 (a) Is the amount of water in the tank increasing at time t = 15? Why or why not? (b) To the nearest whole number, how many gallons of water are in the tank at time t = 18? (c) At what time t, for t 18, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion. (d) For t > 18, 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 t = 15 since W( 15) R( 15) = 11.9 <. 1 : answer with reason 18 (b) 1 + ( W() t R() t ) dt = gallons 3 : 1 : limits 1 : integrand 1 : answer (c) W() t R() t = t =, , t (hours) gallons of water : interior critical points 1 : amount of water is least at 3 : t = or : analysis for absolute minimum The values at the endpoints and the critical points show that the absolute minimum occurs when t = or k (d) Rt () dt= : limits : 1 : equation Copyright 5 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 3 3

10 AP CALCULUS AB 4 SCORING GUIDELINES Question 1 Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t () 8 4sin ( ) Ft = + for t 3, where F() t is measured in cars per minute and t is measured in minutes. (a) To the nearest whole number, how many cars pass through the intersection over the 3-minute period? (b) Is the traffic flow increasing or decreasing at t = 7? Give a reason for your answer. (c) What is the average value of the traffic flow over the time interval 1 t 15? Indicate units of measure. (d) What is the average rate of change of the traffic flow over the time interval 1 t 15? Indicate units of measure. 3 (a) Ft () dt= 474 cars 3 : 1 : limits 1 : integrand 1 : answer (b) F ( 7) = 1.87 or Since F ( 7) <, the traffic flow is decreasing at t = 7. 1 : answer with reason 1 15 (c) () cars min 5 Ft dt= 1 3 : 1 : limits 1 : integrand 1 : answer (d) F( 15) F( 1) 15 1 = or cars min 1 : answer Units of cars min in (c) and cars min in (d) 1 : units in (c) and (d) Copyright 4 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 9

11 AP CALCULUS AB 4 SCORING GUIDELINES (Form B) Question For t 31, the rate of change of the number of mosquitoes on Tropical Island at time t days is t Rt = 5 tcos mosquitoes per day. There are 1 mosquitoes on Tropical Island at 5 modeled by () ( ) time t =. (a) Show that the number of mosquitoes is increasing at time t = 6. (b) At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer. (c) According to the model, how many mosquitoes will be on the island at time t = 31? Round your answer to the nearest whole number. (d) To the nearest whole number, what is the maximum number of mosquitoes for t 31? Show the analysis that leads to your conclusion. (a) Since R ( 6) = >, the number of mosquitoes is increasing at t = 6. 1 : shows that R ( 6) > (b) R ( 6) = Since R ( 6) <, the number of mosquitoes is increasing at a decreasing rate at t = 6. : 1 : considers R ( 6) 1 : answer with reason 31 (c) 1 + Rt () dt= To the nearest whole number, there are 964 mosquitoes. 1 : integral : 1 : answer (d) Rt () = when t =, t =.5π, or t = 7.5π Rt () > on < t <.5π Rt () < on.5π < t < 7.5π Rt () > on 7.5π < t < 31 The absolute maximum number of mosquitoes occurs at t =.5π or at t = 31..5π 1 + Rt () dt= , There are 964 mosquitoes at t = 31, so the maximum number of mosquitoes is 139, to the nearest whole number. 4 : : absolute maximum value 1 : integral 1 : answer : analysis 1 : computes interior critical points 1 : completes analysis Copyright 4 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 8 3

12 AP CALCULUS AB SCORING GUIDELINES (Form B) Question J 6DAK>AHBC=IJ B=FKJ=JE==A?D=CAI=JJDAH=JA= J Г! A Г C=IFAH@=OMDAHAJEIA=IKHA@E@=OI6DAHA=HA#C=IBJDAFKJ=JEJDA=A=J JEAJ6DA=AEI?IE@AHA@J>AI=BAMDAEJ?J=EI"C=IHAIIBFKJ=J = 1IJDA=KJBFKJ=JE?HA=IEC=JJEAJ'9DOHMDOJ >.HMD=JL=KABJMEJDAK>AHBC=IBFKJ=J>A=JEJIEEKKIJEBOOKH =IMAH? 1IJDA=AI=BAMDAJDAK>AHBC=IBFKJ=JEI=JEJIEEKKIJEBOOKH )ELAIJEC=JHKIAIJDAJ=CAJEA=FFHNE=JEJJ =JJ=I=@ABHJDA = > =KJBFKJ=JEJDA=A)JMD=JJEAJ@AIJDEI@AFHA@E?JJD=JJDA=A >A?AII=BA = $ ' Г! Г$"$ A Г IJDA=KJEIJE?HA=IEC=JJDEIJEA = Г! J J A Г J #!!%" = J EIAC=JELABHJ #! BHJ J #! #! =@FIEJELA 6DAHABHAJDAHAEI=EEK=J!%" Г J?!%" # = Г! Г 6DA=AME>A?AI=BA MDAJDA=KJ@A?HA=IAI>O)EA=H@A FHA@E?JIJDEIMED=FFAMDAJ# Copyright by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 4 3 =IMAHMEJDHA=I IAJI= J! ILAIBHJ KIJEBE?=JE EJACH=@ EEJI!??KIEMEJDHA=I >=IA@EJACH=B= J IFABJ=CAJEA =IMAH

1998 Calculus AB Scoring Guidelines

41 Velocity (feet per second) v(t) 9 8 7 6 5 4 1 O 1998 Calculus AB Scoring Guidelines 5 1 15 5 5 4 45 5 Time (seconds) t t v(t) (seconds) (feet per second) 5 1 1 15 55 5 7 78 5 81 4 75 45 6 5 7. The graph