SCORING GUIDELINES (Form B) Quesion A ank conains 15 gallons of heaing oil a ime =. During he ime inerval 1 hours, heaing oil is pumped ino he ank a he rae 1 H ( ) = + ( 1 + ln( + 1) ) gallons per hour. During he same ime inerval, heaing oil is removed from he ank a he rae R( ) = 1 sin gallons per hour. 47 (a) How many gallons of heaing oil are pumped ino he ank during he ime inerval 1 hours? (b) Is he level of heaing oil in he ank rising or falling a ime = 6 hours? Give a reason for your answer. (c) How many gallons of heaing oil are in he ank a ime = 1 hours? (d) A wha ime, for 1, is he volume of heaing oil in he ank he leas? Show he analysis ha leads o your conclusion. (a) 1 Hd () = 7.57 or 7.571 : 1 : inegral (b) H(6)(6) R =.94, so he level of heaing oil is falling a = 6. wih reason 1 (c) 15 + ( H ( )( R) d = 1.5 or 1.6 : 1 : limis 1 : inegrand (d) The absolue minimum occurs a a criical poin or an endpoin. H ()() R = when = 4.79 and = 11.18. The volume increases unil = 4.79, hen decreases unil = 11.18, hen increases, so he absolue minimum will be a = or a 1 : ses H ( )( R) = 1 : volume is leas a : = 11.18 1 : analysis for absolue minimum = 11.18. 11.18 15 + ( H ()() R) d = 1.78 Since he volume is 15 a =, he volume is leas a = 11.18. Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com.
4 SCORING GUIDELINES (Form B) Quesion For 1, he rae of change of he number of mosquioes on Tropical Island a ime days is R = 5 cos mosquioes per day. There are 1 mosquioes on Tropical Island a 5 modeled by () ( ) ime =. (a) Show ha he number of mosquioes is increasing a ime = 6. (b) A ime = 6, is he number of mosquioes increasing a an increasing rae, or is he number of mosquioes increasing a a decreasing rae? Give a reason for your answer. (c) According o he model, how many mosquioes will be on he island a ime = 1? Round your answer o he neares whole number. (d) To he neares whole number, wha is he maximum number of mosquioes for 1? Show he analysis ha leads o your conclusion. (a) Since R ( 6) = 4.48 >, he number of mosquioes is increasing a = 6. 1 : shows ha R ( 6) > (b) R ( 6) = 1.91 Since R ( 6) <, he number of mosquioes is increasing a a decreasing rae a = 6. : 1 : considers R ( 6) wih reason 1 (c) 1 + R () d= 964.5 To he neares whole number, here are 964 mosquioes. 1 : inegral : (d) R () = when =, =.5π, or = 7.5π R () > on < <.5π R () < on.5π < < 7.5π R () > on 7.5π < < 1 The absolue maximum number of mosquioes occurs a =.5π or a = 1..5π 1 + R () d= 19.57, There are 964 mosquioes a = 1, so he maximum number of mosquioes is 19, o he neares whole number. 4 : : absolue maximum value 1 : inegral : analysis 1 : compues inerior criical poins 1 : complees analysis Copyrigh 4 by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for AP sudens and parens).
5 SCORING GUIDELINES (Form B) Quesion A waer ank a Camp Newon holds 1 gallons of waer a ime =. During he ime inerval 18 hours, waer is pumped ino he ank a he rae () 95 sin ( ) W = gallons per hour. 6 During he same ime inerval, waer is removed from he ank a he rae () 75sin ( ) R = gallons per hour. (a) Is he amoun of waer in he ank increasing a ime = 15? Why or why no? (b) To he neares whole number, how many gallons of waer are in he ank a ime = 18? (c) A wha ime, for 18, is he amoun of waer in he ank a an absolue minimum? Show he work ha leads o your conclusion. (d) For > 18, no waer is pumped ino he ank, bu waer coninues o be removed a he rae R() unil he ank becomes empy. Le k be he ime a which he ank becomes empy. Wrie, bu do no solve, an equaion involving an inegral expression ha can be used o find he value of k. (a) No; he amoun of waer is no increasing a = 15 since W( 15) R( 15) = 11.9 <. wih reason 18 (b) 1 + ( W() R() ) d = 19.788 11 gallons : 1 : limis 1 : inegrand (c) W() R() = =, 6.4948, 1.9748 (hours) gallons of waer 1 6.495 55 1.975 1697 18 11 1 : inerior criical poins 1 : amoun of waer is leas a : = 6.494 or 6.495 1 : analysis for absolue minimum The values a he endpoins and he criical poins show ha he absolue minimum occurs when = 6.494 or 6.495. k (d) R () d= 11 18 1 : limis : 1 : equaion Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for AP sudens and parens).
7 SCORING GUIDELINES (Form B) Quesion The wind chill is he emperaure, in degrees Fahrenhei ( F, ) a human feels based on he air emperaure, in degrees Fahrenhei, and he wind velociy v, in miles per hour ( mph ). If he air emperaure is F, hen he.16 wind chill is given by W( v) = 55.6.1v and is valid for 5 v 6. (a) Find W ( ). Using correc unis, explain he meaning of W ( ) in erms of he wind chill. (b) Find he average rae of change of W over he inerval 5 v 6. Find he value of v a which he insananeous rae of change of W is equal o he average rae of change of W over he inerval 5 v 6. (c) Over he ime inerval 4 hours, he air emperaure is a consan F. A ime =, he wind velociy is v = mph. If he wind velociy increases a a consan rae of 5 mph per hour, wha is he rae of change of he wind chill wih respec o ime a = hours? Indicae unis of measure. (a).84 W ( ) =.1.16 =.85 or.86 When v = mph, he wind chill is decreasing a.86 F mph. : { 1 : value 1 : explanaion (b) The average rae of change of W over he inerval W( 6) W( 5) 5 v 6 is =.5 or.54. 6 5 W( 6) W( 5) W ( v) = when v =.11. 6 5 1 : average rae of change : 1 : W ( v) = average rae of change 1 : value of v dw dw dv = = 5 5 =.89 F hr d dv d (c) ( ) W ( ) = = OR W = 55.6.1( + 5).16 dw =.89 F hr d = dv 1 : = 5 d 1 : uses v( ) = 5, : or uses v () = + 5 Unis of F mph in (a) and F hr in (c) 1 : unis in (a) and (c) 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for sudens and parens).
For ime AP CALCULUS AB 8 SCORING GUIDELINES (Form B) 1 hours, le r () 1( 1 e ) Quesion = represen he speed, in kilomeers per hour, a which a car ravels along a sraigh road. The number of liers of gasoline used by he car o ravel x kilomeers is g x =.5x 1 e x. modeled by ( ) ( ) (a) How many kilomeers does he car ravel during he firs hours? (b) Find he rae of change wih respec o ime of he number of liers of gasoline used by he car when = hours. Indicae unis of measure. (c) How many liers of gasoline have been used by he car when i reaches a speed of 8 kilomeers per hour? kilomeers : { 1 : inegral (a) r() d = 6.7 dg dg dx dx (b) = ; = r () d dx d d dg dg r( ) d = dx = x= 6.7 = (.5)( 1) = 6 liers hour : uses chain rule : { wih unis (c) Le T be he ime a which he car s speed reaches 8 kilomeers per hour. Then, rt ( ) = 8 or T =.145 hours. 4 : 1 : equaion r () = 8 : disance inegral A ime T, he car has gone T xt ( ) = r( ) d= 1.79497 kilomeers and has consumed g( x( T )) =.57 liers of gasoline. 8 The College Board. All righs reserved. Visi he College Board on he Web: www.collegeboard.com.
9 SCORING GUIDELINES (Form B) Quesion 1 A a cerain heigh, a ree runk has a circular cross secion. The radius R() of ha cross secion grows a a rae modeled by he funcion dr = 1 ( + sin ( )) cenimeers per year d 16 for, where ime is measured in years. A ime =, he radius is 6 cenimeers. The area of he cross secion a ime is denoed by A (). (a) Wrie an expression, involving an inegral, for he radius R() for. Use your expression o find R (. ) (b) Find he rae a which he cross-secional area A() is increasing a ime = years. Indicae unis of measure. (c) Evaluae A () d. Using appropriae unis, inerpre he meaning of ha inegral in erms of crosssecional area. 1 (a) () = 6 + ( + sin( )) R 16 x dx R ( ) = 6.61 or 6.611 1 : inegral : 1 : expression for R() 1 : R( ) (b) A () = π ( R ()) A () = π R() R () A ( ) = 8.858 cm year 1 : expression for A () : 1 : expression for A () wih unis (c) A () d = A ( ) A ( ) = 4. or 4.1 From ime = o = years, he crosssecional area grows by 4.1 square cenimeers. 1 : uses Fundamenal Theorem of Calculus 1 : value of A () d : 1 : meaning of A () d 9 The College Board. All righs reserved. Visi he College Board on he Web: www.collegeboard.com.
11 SCORING GUIDELINES (Form B) Quesion 1 A cylindrical can of radius 1 millimeers is used o measure rainfall in Sormville. The can is iniially empy, and rain eners he can during a 6-day period. The heigh of waer in he can is modeled by he funcion S, where S () is measured in millimeers and is measured in days for 6. The rae a which he heigh of he waer is rising in he can is given by S ( ) = sin(.) + 1.5. (a) According o he model, wha is he heigh of he waer in he can a he end of he 6-day period? (b) According o he model, wha is he average rae of change in he heigh of waer in he can over he 6-day period? Show he compuaions ha lead o your answer. Indicae unis of measure. (c) Assuming no evaporaion occurs, a wha rae is he volume of waer in he can changing a ime = 7? Indicae unis of measure. (d) During he same 6-day period, rain on Monsoon Mounain accumulaes in a can idenical o he one in Sormville. The heigh of he waer in he can on Monsoon Mounain is modeled by he funcion M, where 1 M () = ( + ). The heigh M ( ) is measured in millimeers, and is measured in days 4 for 6. Le D ( ) = M ( ) S ( ). Apply he Inermediae Value Theorem o he funcion D on he inerval 6 o jusify ha here exiss a ime, < < 6, a which he heighs of waer in he wo cans are changing a he same rae. 6 (a) S( 6) = S ( ) d = 171.81 mm : 1 : limis 1 : inegrand (b) S( 6) S( ) 6 =.86 or.864 mm day (c) V() = 1π S() V ( 7) = 1π S ( 7) = 6.18 The volume of waer in he can is increasing a a rae of 6.18 mm day. 1 : relaionship beween V and S : { (d) D ( ) =.675 < and D ( 6) = 69.77 > Because D is coninuous, he Inermediae Value Theorem implies ha here is a ime, < < 6, a which D ( ) =. A his ime, he heighs of waer in he wo cans are changing a he same rae. 1 : considers D( ) and D( 6) : 1 : jusificaion 1 : unis in (b) or (c) 11 The College Board. Visi he College Board on he Web: www.collegeboard.org.