AP CALCULUS AB 2017 SCORING GUIDELINES

Transcription

1 AP CALCULUS AB 17 SCORING GUIDELINES

2 /CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured in hours. Waer is removed from he ank a a rae modeled by R ( ) liers per hour, where R is differeniable and decreasing on 8. Seleced values of R ( ) are shown in he able above. A ime =, here are 5, liers of waer in he ank. (a) Esimae R (. ) Show he work ha leads o your answer. Indicae unis of measure. (b) Use a lef Riemann sum wih he four subinervals indicaed by he able o esimae he oal amoun of waer removed from he ank during he 8 hours. Is his an overesimae or an underesimae of he oal amoun of waer removed? Give a reason for your answer. (c) Use your answer from par (b) o find an esimae of he oal amoun of waer in he ank, o he neares lier, a he end of 8 hours. (d) For 8, is here a ime when he rae a which waer is pumped ino he ank is he same as he rae a which waer is removed from he ank? Explain why or why no. R( ) R( 1) { (a) R ( ) = = 1 liers/hr : 1 : esimae : unis (b) The oal amoun of waer removed is given by R( ) d. 8 R ( ) d 1 R( ) + R( 1) + R( ) + R( 6) = 1( 14) + ( 119) + ( 95) + ( 74) = 85 liers 8 1 : lef Riemann sum : 1 : esimae 1 : overesimae wih reason This is an overesimae since R is a decreasing funcion. (c) Toal 5 W ( ) d { 1 : inegral = liers : 1 : esimae (d) W( ) R( ) >, W( 8) R( 8) <, and W ( ) R ( ) is coninuous. { 1 : considers W ( ) R ( ) : wih explanaion Therefore, he Inermediae Value Theorem guaranees a leas one ime, < < 8, for which W ( ) R ( ) =, or W ( ) = R ( ). For his value of, he rae a which waer is pumped ino he ank is he same as he rae a which waer is removed from he ank. 16 The College Board. Visi he College Board on he Web:

3 /CALCULUS BC 15 SCORING GUIDELINES Quesion 1 The rae a which rainwaer flows ino a drainpipe is modeled by he funcion R, where R ( ) = sin 5 cubic fee per hour, is measured in hours, and 8. The pipe is parially blocked, allowing waer o drain ou he oher end of he pipe a a rae modeled by D ( ) = cubic fee per hour, for 8. There are cubic fee of waer in he pipe a ime =. (a) How many cubic fee of rainwaer flow ino he pipe during he 8-hour ime inerval 8? (b) Is he amoun of waer in he pipe increasing or decreasing a ime = hours? Give a reason for your answer. (c) A wha ime, 8, is he amoun of waer in he pipe a a minimum? Jusify your answer. (d) The pipe can hold 5 cubic fee of waer before overflowing. For > 8, waer coninues o flow ino and ou of he pipe a he given raes unil he pipe begins o overflow. Wrie, bu do no solve, an equaion involving one or more inegrals ha gives he ime w when he pipe will begin o overflow. 8 : { 1 : inegrand (a) R( ) d = (b) R( ) D( ) =.16 < Since R( ) < D(, ) he amoun of waer in he pipe is decreasing a ime = hours. (c) The amoun of waer in he pipe a ime, 8, is + [ R( x) D( x) ] dx. 1 : considers R( ) and D( ) : and reason 1 : considers R ( ) D ( ) = : 1 : jusificaion R ( ) D ( ) = =, Amoun of waer in he pipe The amoun of waer in he pipe is a minimum a ime =.7 (or.71) hours. w (d) + [ R ( ) D ( )] d = 5 : { 1 : inegral 1 : equaion 15 The College Board. Visi he College Board on he Web:

4 /CALCULUS BC 14 SCORING GUIDELINES Quesion 1 Grass clippings are placed in a bin, where hey decompose. For, he amoun of grass clippings remaining in he bin is modeled by A ( ) = 6.687(.91 ), where A () is measured in pounds and is measured in days. (a) Find he average rae of change of A ( ) over he inerval. Indicae unis of measure. (b) Find he value of A ( 15 ). Using correc unis, inerpre he meaning of he value in he conex of he problem. (c) Find he ime for which he amoun of grass clippings in he bin is equal o he average amoun of grass clippings in he bin over he inerval. (d) For >, L ( ), he linear approximaion o A a =, is a beer model for he amoun of grass clippings remaining in he bin. Use L ( ) o predic he ime a which here will be.5 pound of grass clippings remaining in he bin. Show he work ha leads o your answer. (a) A( ) A( ) =.197 (or.196) lbs/day wih unis (b) A ( 15 ) = (or.16) The amoun of grass clippings in he bin is decreasing a a rae of.164 (or.16) lbs/day a ime = 15 days. 1 : A ( 15) : 1 : inerpreaion (c) 1 A ( ) = A ( ) d (or ) = 1 1 : A( ) d : (d) L ( ) = A( ) + A ( ) ( ) A ( ) = A( ) =.7898 : expression for L ( ) 4 : 1 : L ( ) =.5 L ( ) =.5 = The College Board. Visi he College Board on he Web:

5 1 SCORING GUIDELINES Quesion 1 On a cerain workday, he rae, in ons per hour, a which unprocessed gravel arrives a a gravel processing plan is modeled by G ( ) = cos, 18 where is measured in hours and 8. A he beginning of he workday ( =, ) he plan has 5 ons of unprocessed gravel. During he hours of operaion, 8, he plan processes gravel a a consan rae of 1 ons per hour. (a) Find G ( 5. ) Using correc unis, inerpre your answer in he conex of he problem. (b) Find he oal amoun of unprocessed gravel ha arrives a he plan during he hours of operaion on his workday. (c) Is he amoun of unprocessed gravel a he plan increasing or decreasing a ime = 5 hours? Show he work ha leads o your answer. (d) Wha is he maximum amoun of unprocessed gravel a he plan during he hours of operaion on his workday? Jusify your answer. (a) G ( 5) = (or ) The rae a which gravel is arriving is decreasing by (or 4.587) ons per hour per hour a ime = 5 hours. 1 : G ( 5) : 1 : inerpreaion wih unis 8 ons : { 1 : inegral (b) G( ) d = (c) G ( 5) = < 1 A ime = 5, he rae a which unprocessed gravel is arriving is less han he rae a which i is being processed. Therefore, he amoun of unprocessed gravel a he plan is decreasing a ime = 5. (d) The amoun of unprocessed gravel a ime is given by A ( ) = 5 + ( Gs ( ) 1 ) ds. A ( ) = G ( ) 1 = = : compares G( 5 ) o 1 : 1 : conclusion : 1 : considers A ( ) = 1 : jusificaion A ( ) The maximum amoun of unprocessed gravel a he plan during his workday is ons. 1 The College Board. Visi he College Board on he Web:

6 11 SCORING GUIDELINES (Form B) Quesion A 1,-lier ank of waer is filled o capaciy. A ime =, waer begins o drain ou of he ank a a rae modeled by r (), measured in liers per hour, where r is given by he piecewise-defined funcion 6 for 5 r () = +. 1e for > 5 (a) Is r coninuous a = 5? Show he work ha leads o your answer. (b) Find he average rae a which waer is draining from he ank beween ime = and ime = 8 hours. (c) Find r (. ) Using correc unis, explain he meaning of ha value in he conex of his problem. (d) Wrie, bu do no solve, an equaion involving an inegral o find he ime A when he amoun of waer in he ank is 9 liers. (a) lim r () ( ) = lim = 75 = r( 5) () ( e ) lim r = lim 1 = : conclusion wih analysis Because he lef-hand and righ-hand limis are no equal, r is no coninuous a = 5. 5 (b) r () d= 1 8 d e d = 58.5 or 58.5 : 1 : inegrand 1 : limis and consan (c) r ( ) = 5 The rae a which waer is draining ou of he ank a ime = hours is increasing a 5 liers hour. : 1 : r ( ) 1 : meaning of r ( ) A (d) 1, r () d= 9 : { 1 : inegral 1 : equaion 11 The College Board. Visi he College Board on he Web:

7 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(.) (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 = 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) = > 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:

8 1 SCORING GUIDELINES Quesion 1 There is no snow on Jane s driveway when snow begins o fall a midnigh. From midnigh o 9 A.M., snow cos accumulaes on he driveway a a rae modeled by f() = 7e cubic fee per hour, where is measured in hours since midnigh. Jane sars removing snow a 6 A.M. ( = 6. ) The rae g (), in cubic fee per hour, a which Jane removes snow from he driveway a ime hours afer midnigh is modeled by for < 6 g () = 15 for 6 < 7 18 for 7 9. (a) How many cubic fee of snow have accumulaed on he driveway by 6 A.M.? (b) Find he rae of change of he volume of snow on he driveway a 8 A.M. (c) Le h () represen he oal amoun of snow, in cubic fee, ha Jane has removed from he driveway a ime hours afer midnigh. Express h as a piecewise-defined funcion wih domain 9. (d) How many cubic fee of snow are on he driveway a 9 A.M.? 6 or cubic fee : { 1 : inegral (a) f() d = (b) Rae of change is f( 8) g( 8) = or cubic fee per hour. (c) h ( ) = For < 6, h () = h( ) + gs ( ) ds= + ds=. For 6 7, < h () = h( 6) + gs ( ) ds= + 15 ds= 15( 6 ). For 7 9, 6 6 < h () = h( 7) + g( s) ds = ds = ( 7 ). 7 7 for 6 Thus, h () = 15( 6) for 6 < ( 7) for 7 < 9 : 1 : h () for 6 1 : h () for 6 < 7 1 : h () for 7 < 9 1 : inegral 9 (d) Amoun of snow is f() d h( 9) = 6.4 or 6.5 cubic fee. : 1 : h( 9) 1 The College Board. Visi he College Board on he Web:

9 9 SCORING GUIDELINES (Form B) Quesion A sorm washed away sand from a beach, causing he edge of he waer o ge closer o a nearby road. The rae a which he disance beween he road and he edge of he waer was changing during he sorm is modeled by f() = + cos meers per hour, hours afer he sorm began. The edge of he waer was 5 meers from he road when he sorm began, and he sorm lased 5 hours. The derivaive of f () 1 is f () = sin. (a) Wha was he disance beween he road and he edge of he waer a he end of he sorm? (b) Using correc unis, inerpre he value f ( 4) = 1.7 in erms of he disance beween he road and he edge of he waer. (c) A wha ime during he 5 hours of he sorm was he disance beween he road and he edge of he waer decreasing mos rapidly? Jusify your answer. (d) Afer he sorm, a machine pumped sand back ono he beach so ha he disance beween he road and he edge of he waer was growing a a rae of g( p ) meers per day, where p is he number of days since pumping began. Wrie an equaion involving an inegral expression whose soluion would give he number of days ha sand mus be pumped o resore he original disance beween he road and he edge of he waer. 5 or meers : { 1 : inegral (a) 5 + f() d = (b) Four hours afer he sorm began, he rae of change of he disance beween he road and he edge of he waer is increasing a a rae of 1.7 meers hours. (c) f () = when = and =.848 The minimum of f for 5 may occur a,.66187,.848, or 5. f ( ) = f (.66187) = f (.848) =.696 f ( 5) = : inerpreaion of f ( 4) : 1 : unis : 1 : considers f () = 1 : jusificaion The disance beween he road and he edge of he waer was decreasing mos rapidly a ime =.84 hours afer he sorm began. 5 x : { (d) f () d = g( p) dp 1 : inegral of g 9 The College Board. All righs reserved. Visi he College Board on he Web:

10 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 : inegral : 1 : expression for R() 1 : R( ) (b) A () = π ( R ()) A () = π R() R () A ( ) = 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:

11 9 SCORING GUIDELINES Quesion Mighy Cable Company manufacures cable ha sells for $1 per meer. For a cable of fixed lengh, he cos of producing a porion of he cable varies wih is disance from he beginning of he cable. Mighy repors ha he cos o produce a porion of a cable ha is x meers from he beginning of he cable is 6 x dollars per meer. (Noe: Profi is defined o be he difference beween he amoun of money received by he company for selling he cable and he company s cos of producing he cable.) (a) Find Mighy s profi on he sale of a 5-meer cable. (b) Using correc unis, explain he meaning of 6 x dx in he conex of his problem. 5 (c) Wrie an expression, involving an inegral, ha represens Mighy s profi on he sale of a cable ha is k meers long. (d) Find he maximum profi ha Mighy could earn on he sale of one cable. Jusify your answer. (a) Profi = 5 = dollars : { 1 : inegral xdx 5 (b) 6 x dx is he difference in cos, in dollars, of producing a 5 cable of lengh meers and a cable of lengh 5 meers. wih unis (c) Profi 1 k k 6 x dx = dollars : { 1 : inegral 1 : expression (d) Le P( k ) be he profi for a cable of lengh k. P ( k) = 1 6 k = when k = 4. This is he only criical poin for P, and P changes from posiive o negaive a k = 4. Therefore, he maximum profi is P ( 4) = 16, dollars. 4 : 1 : P ( k) = 1 : k = 4 1 : jusificaion 9 The College Board. All righs reserved. Visi he College Board on he Web:

12 9 SCORING GUIDELINES Quesion The rae a which people ener an audiorium for a rock concer is modeled by he funcion R given by R() = for hours; R() is measured in people per hour. No one is in he audiorium a ime =, when he doors open. The doors close and he concer begins a ime =. (a) How many people are in he audiorium when he concer begins? (b) Find he ime when he rae a which people ener he audiorium is a maximum. Jusify your answer. (c) The oal wai ime for all he people in he audiorium is found by adding he ime each person wais, saring a he ime he person eners he audiorium and ending when he concer begins. The funcion w models he oal wai ime for all he people who ener he audiorium before ime. The derivaive of w is given by w () = ( ) R(). Find w( ) w( 1 ), he oal wai ime for hose who ener he audiorium afer ime = 1. (d) On average, how long does a person wai in he audiorium for he concer o begin? Consider all people who ener he audiorium afer he doors open, and use he model for oal wai ime from par (c). people : { 1 : inegral (a) R () d= 98 (b) R () = when = and = The maximum rae may occur a, a = 1.696, or. R ( ) = Ra ( ) = R ( ) = 1 : 1 : considers R () = 1 : inerior criical poin and jusificaion The maximum rae occurs when = 1.6 or 1.6. (c) w( ) w() 1 = w () d = ( ) R() d = The oal wai ime for hose who ener he audiorium afer ime = 1 is 87.5 hours. : { 1 : inegral 1 1 (d) w( ) = ( ) R( ) d = On average, a person wais.775 or.776 hour. : { 1 : inegral 9 The College Board. All righs reserved. Visi he College Board on he Web:

13 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= 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:

14 8 SCORING GUIDELINES Quesion Oil is leaking from a pipeline on he surface of a lake and forms an oil slick whose volume increases a a consan rae of cubic cenimeers per minue. The oil slick akes he form of a righ circular cylinder wih boh is radius and heigh changing wih ime. (Noe: The volume V of a righ circular cylinder wih radius r and heigh h is given by V = π r h. ) (a) A he insan when he radius of he oil slick is 1 cenimeers and he heigh is.5 cenimeer, he radius is increasing a he rae of.5 cenimeers per minue. A his insan, wha is he rae of change of he heigh of he oil slick wih respec o ime, in cenimeers per minue? (b) A recovery device arrives on he scene and begins removing oil. The rae a which oil is removed is R() = 4 cubic cenimeers per minue, where is he ime in minues since he device began working. Oil coninues o leak a he rae of cubic cenimeers per minue. Find he ime when he oil slick reaches is maximum volume. Jusify your answer. (c) By he ime he recovery device began removing oil, 6, cubic cenimeers of oil had already leaked. Wrie, bu do no evaluae, an expression involving an inegral ha gives he volume of oil a he ime found in par (b). (a) When r = 1 cm and h =.5 cm, and dr d =.5 cm min. dv dr dh = πr h + πr d d d = π( 1)(.5)(.5) + π( 1) dh d =.8 or.9 cm min dv d dh d = cm min dv dr 1 : = and =.5 d d 4 : dv : expression for d dv dv (b) = R(), so = when R () =. d d This occurs when = 5 minues. dv dv Since > for < < 5 and < for > 5, d d he oil slick reaches is maximum volume 5 minues afer he device begins working. : 1 : R () = 1 : jusificaion (c) The volume of oil, in cm, in he slick a ime = 5 minues 5 is given by 6, + ( R() ) d. 1 : limis and iniial condiion : { 1 : inegrand 8 The College Board. All righs reserved. Visi he College Board on he Web:

15 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) = v 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 W( 6) W( 5) W ( v) = when v = : 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 = ( + 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 (for sudens and parens).

16 7 SCORING GUIDELINES Quesion The amoun of waer in a sorage ank, in gallons, is modeled by a coninuous funcion on he ime inerval 7, where is measured in hours. In his model, raes are given as follows: (i) The rae a which waer eners he ank is f () = 1 sin ( ) gallons per hour for 7. (ii) The rae a which waer leaves he ank is 5 for < g () = gallons per hour. for < 7 The graphs of f and g, which inersec a = and = 5.76, are shown in he figure above. A ime =, he amoun of waer in he ank is 5 gallons. (a) How many gallons of waer ener he ank during he ime inerval 7? Round your answer o he neares gallon. (b) For 7, find he ime inervals during which he amoun of waer in he ank is decreasing. Give a reason for each answer. (c) For 7, a wha ime is he amoun of waer in he ank greaes? To he neares gallon, compue he amoun of waer a his ime. Jusify your answer. 7 gallons : { 1 : inegral (a) f() d 864 (b) The amoun of waer in he ank is decreasing on he inervals and 5.76 because f () < g() for < and < < (c) Since f () g() changes sign from posiive o negaive only a =, he candidaes for he absolue maximum are a =,, and 7. (hours) gallons of waer f() d 5( ) = f() d ( 4) = : { 1 : inervals 1 : reason 1 : idenifies = as a candidae 1 : inegrand 5 : 1 : amoun of waer a = 1 : amoun of waer a = 7 1 : conclusion The amoun of waer in he ank is greaes a hours. A ha ime, he amoun of waer in he ank, rounded o he neares gallon, is 517 gallons. 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

17 6 SCORING GUIDELINES Quesion A an inersecion in Thomasville, Oregon, cars urn lef a he rae L () = 6 sin ( ) cars per hour over he ime inerval 18 hours. The graph of y = L() is shown above. (a) To he neares whole number, find he oal number of cars urning lef a he inersecion over he ime inerval 18 hours. (b) Traffic engineers will consider urn resricions when L () 15 cars per hour. Find all values of for which L () 15 and compue he average value of L over his ime inerval. Indicae unis of measure. (c) Traffic engineers will insall a signal if here is any wo-hour ime inerval during which he produc of he oal number of cars urning lef and he oal number of oncoming cars raveling sraigh hrough he inersecion is greaer han,. In every wo-hour ime inerval, 5 oncoming cars ravel sraigh hrough he inersecion. Does his inersecion require a raffic signal? Explain he reasoning ha leads o your conclusion. 18 (a) L () d 1658 cars (b) L () = 15 when = 1.481, Le R = and S = L () 15 for in he inerval [ R, S ] 1 S L () d= S R cars per hour R (c) For he produc o exceed,, he number of cars urning lef in a wo-hour inerval mus be greaer han OR L () d= > 4 The number of cars urning lef will be greaer han 4 on a wo-hour inerval if L () on ha inerval. L () on any wo-hour subinerval of [ 1.54, ]. : { 1 : seup : 1 : -inerval when L() 15 1 : average value inegral wih unis 1 : considers 4 cars 1 : valid inerval [ h, h ] + 4 : h+ 1 : value of L () d h and explanaion 4 : OR 1 : considers cars per hour 1 : solves L () 1 : discusses hour inerval and explanaion Yes, a raffic signal is required. 6 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

18 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 = gallons : 1 : limis 1 : inegrand (c) W() R() = =, , (hours) gallons of waer : inerior criical poins 1 : amoun of waer is leas a : = or : analysis for absolue minimum The values a he endpoins and he criical poins show ha he absolue minimum occurs when = or k (d) R () d= : limis : 1 : equaion Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

19 5 SCORING GUIDELINES Quesion The ide removes sand from Sandy Poin Beach a a rae modeled by he funcion R, given by 4π () = + ( ) R 5sin. 5 A pumping saion adds sand o he beach a a rae modeled by he funcion S, given by 15 S () =. 1 + Boh R() and S () have unis of cubic yards per hour and is measured in hours for 6. A ime =, he beach conains 5 cubic yards of sand. (a) How much sand will he ide remove from he beach during his 6-hour period? Indicae unis of measure. (b) Wrie an expression for Y (), he oal number of cubic yards of sand on he beach a ime. (c) Find he rae a which he oal amoun of sand on he beach is changing a ime = 4. (d) For 6, a wha ime is he amoun of sand on he beach a minimum? Wha is he minimum value? Jusify your answers. 6 (a) R () d= or yd : { 1 : inegral wih unis (b) Y( ) = 5 + ( S( x) R( x) ) dx : 1 : inegrand 1 : limis (c) Y () = S() R() Y ( 4) = S( 4) R( 4) = 1.98 or 1.99 yd hr (d) Y () = when S () R () =. The only value in [, 6 ] o saisfy S () = R () is a = : 1 : ses Y () = 1 : criical -value wih jusificaion Y() 5 a The amoun of sand is a minimum when = or hours. The minimum value is cubic yards. Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

20 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= 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 (for AP sudens and parens).

21 4 SCORING GUIDELINES Quesion 1 Traffic flow is defined as he rae a which cars pass hrough an inersecion, measured in cars per minue. The raffic flow a a paricular inersecion is modeled by he funcion F defined by () 8 4sin ( ) F = + for, where F() is measured in cars per minue and is measured in minues. (a) To he neares whole number, how many cars pass hrough he inersecion over he -minue period? (b) Is he raffic flow increasing or decreasing a = 7? Give a reason for your answer. (c) Wha is he average value of he raffic flow over he ime inerval 1 15? Indicae unis of measure. (d) Wha is he average rae of change of he raffic flow over he ime inerval 1 15? Indicae unis of measure. (a) F () d= 474 cars : 1 : limis 1 : inegrand (b) F ( 7) = 1.87 or 1.87 Since F ( 7) <, he raffic flow is decreasing a = 7. wih reason 1 15 (c) () cars min 5 F d= 1 : 1 : limis 1 : inegrand (d) F( 15) F( 1) 15 1 = or cars min Unis of cars min in (c) and cars min in (d) 1 : unis in (c) and (d) Copyrigh 4 by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

22 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 : 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 = 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 : = : analysis for absolue minimum = ( H ()() R) d = 1.78 Since he volume is 15 a =, he volume is leas a = Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com.

23 SCORING GUIDELINES (Form B) Quesion 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 =IMAHMEJDHA=I $ = = ' Г! A Г Г$"$ IJDA=KJEIJE?HA=IEC=JJDEIJEA > = J Г! A Г J IAJI= J J #!!%"! ILAIBHJ = J EIAC=JELABHJ #! =@FIEJELA KIJEBE?=JE BHJ #! 6DAHABHAJDAHAEI=EEK=J J #!!%" Г J?!%" # Г! EJACH=@ EEJI!#""IJDA=AEII=BA!??KIEMEJDHA=I >=IA@EJACH=B= = Г! Г 6DA=AME>A?AI=BA IFABJ=CAJEA MDAJDA=KJ@A?HA=IAI>O)EA=H@A =IMAH FHA@E?JIJDEIMED=FFAMDAJ# Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

24 SCORING GUIDELINES Quesion - J #$ J Г " J $ J '&' J Г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Г% Г!& & 6DAHAMAHA!% #FAFAEJDAF=H=JJ% 6DAK>AHBFAFAEJDAF=HM=I@A?HA=IEC =JJDAH=JAB=FFHNE=JAO!&FAFADH=J = J -J ГJ J#%'"H#%'# Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board. EEJI! EJACH=@ =IMAH IAJKF L=KAB = % A=ECI! A=ECB % A=ECB = % Г EBHABAHA?AJJ % -J ГJ =IMAH

25 !Å#ALCULUSÅ!"n 7ATERÅISÅPUMPEDÅINTOÅANÅUNDERGROUNDÅTANKÅATÅAÅCONSTANTÅRATEÅOFÅÅGALLONSÅPERÅMINUTEÅ7ATERÅLEAKSÅOUT OFÅTHEÅTANKÅATÅTHEÅRATEÅOFÅ T ÅGALLONSÅPERÅMINUTEÅFORÅÅbÅTÅbÅÅMINUTESÅ!TÅTIMEÅTÅÅÅTHEÅTANK CONTAINSÅÅGALLONSÅOFÅWATER A (OWÅMANYÅGALLONSÅOFÅWATERÅLEAKÅOUTÅOFÅTHEÅTANKÅFROMÅTIMEÅTÅÅÅTOÅTÅÅÅMINUTES B (OWÅMANYÅGALLONSÅOFÅWATERÅAREÅINÅTHEÅTANKÅATÅTIMEÅTÅÅÅMINUTES C 7RITEÅANÅEXPRESSIONÅFORÅ!T ÅTHEÅTOTALÅNUMBERÅOFÅGALLONSÅOFÅWATERÅINÅTHEÅTANKÅATÅTIMEÅT D!TÅWHATÅTIMEÅTÅFORÅÅbÅTÅbÅÅISÅTHEÅAMOUNTÅOFÅWATERÅINÅTHEÅTANKÅAÅMAXIMUMÅ*USTIFYÅYOUR ANSWER -ETHODÅ A ÅÅ-ETHODÅÅ T DT T DEFINITEÅINTEGRAL ÅÅÅÅÅÅÅLIMITS ÅÅÅÅÅÅnÅORÅn Å ÅÅÅÅÅÅÅINTEGRAND ÅÅÅÅÅÅ-ETHODÅÅ,T ÅÅGALLONSÅLEAKEDÅINÅFIRSTÅTÅMINUTES ÅÅANSWER D, T, T T # nåorån DT -ETHODÅ, # ÅÅANTIDERIVATIVEÅWITHÅ,T T, Å # ÅÅSOLVESÅFORÅ# ÅUSINGÅÅ, ÅÅANSWER B ÅÅ ÅÅANSWER C -ETHODÅ T!T X DX T X DX nåorån ÅÅÅÅÅÅ-ETHODÅ D! T DT!T T T # ÅÅÅÅÅÅ # #!T T T T -ETHODÅ T Å T X DX nåorån -ETHODÅ ÅÅANTIDERIVATIVEÅWITHÅ# Å ÅÅANSWER D! a T T ÅWHENÅTÅÅ! a T ÅISÅPOSITIVEÅFORÅÅÅTÅÅÅANDÅNEGATIVEÅFOR ÅÅTÅÅÅ4HEREFOREÅTHEREÅISÅAÅMAXIMUM ATÅTÅÅ ÅSETSÅ! a T Å ÅSOLVESÅFORÅT ÅJUSTIFICATION Copyrigh by College Enrance Examinaion Board and Educaional Tesing Service. All righs reserved. AP is a regisered rademark of he College Enrance Examinaion Board.

26 1998 Calculus AB Scoring Guidelines 5. The emperaure ouside a house during a 4-hour period is given by ( ) π F () = 8 1 cos, 4, 1 where F () is measured in degrees Fahrenhei and is measured in hours. (a) Skech he graph of F on he grid below. (b) Find he average emperaure, o he neares degree Fahrenhei, beween = 6 and = 14. (c) An air condiioner cooled he house whenever he ouside emperaure was a or above 78 degrees Fahrenhei. For wha values of was he air condiioner cooling he house? (d) The cos of cooling he house accumulaes a he rae of $.5 per hour for each degree he ouside emperaure exceeds 78 degrees Fahrenhei. Wha was he oal cos, o he neares cen, o cool he house for his 4 hour period? (a) Degrees Fahrenhei : bell shaped graph minimum 7 a =, = 4 only maximum 9 a = 1 only 6 (b) Avg. = (c) = 1 8 ( ) 6 Time in Hours = or F [ 8 1 cos 1 cos 5. or 5.1 ( π 1 ) ( π 1 [ ( )] π 8 1 cos d 1 )] or : inegral 1: limis and 1/(14 6) 1: inegrand 1: answer /1 if inegral no of he form 1 b a b a F () d { 1: inequaliy or equaion 1: soluions wih inerval (d) C = or or 5. ([ ( )] ) π 8 1 cos 78 d 1 =.5( ) = 5.96 $5.1 : inegral 1: limis and.5 1: inegrand 1: answer /1 if inegral no of he form k b a (F () 78) d Copyrigh 1998 College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.