AP CALCULUS AB 17 SCORING GUIDELINES
/CALCULUS BC 15 SCORING GUIDELINES Quesion (minues) v ( ) (meers per minue) 1 4 4 4 15 Johanna jogs along a sraigh pah. For 4, Johanna s velociy is given by a differeniable funcion v. Seleced values of v ( ), where is measured in minues and v ( ) is measured in meers per minue, are given in he able above. (a) Use he daa in he able o esimae he value of v ( 16 ). (b) Using correc unis, explain he meaning of he definie inegral v( ) d in he conex of he problem. 4 Approximae he value of v( ) d using a righ Riemann sum wih he four subinervals indicaed in he able. (c) Bob is riding his bicycle along he same pah. For 1, Bob s velociy is modeled by B ( ) = 6 +, where is measured in minues and B ( ) is measured in meers per minue. Find Bob s acceleraion a ime = 5. (d) Based on he model B from par (c), find Bob s average velociy during he inerval 1. 4 (a) v ( 16) = 5 meers/min 1 : approximaion 1 4 4 (b) v( ) d is he oal disance Johanna jogs, in meers, over he ime inerval 4 minues. 4 v( ) d 1 v( 1) + 8 v( ) + 4 v( 4) + 16 v( 4) = 1 + 8 4 + 4 + 16 15 = 4 + 19 + 88 + 4 = 76 meers 1 : explanaion : 1 : righ Riemann sum 1 : approximaion (c) Bob s acceleraion is B ( ) = 1. B ( 5) = ( 5) 1( 5) = 15 meers/min 1 1 (d) Avg ve ( 6 + ) 15 The College Board. Visi he College Board on he Web: www.collegeboard.org. 1 : uses B ( ) : l = 1 d 1 : inegral 4 1 : 1 : aniderivaive 1 = + 1 4 1 1 = + = 5 meers/ min 1 4
/CALCULUS BC 14 SCORING GUIDELINES Quesion 4 Train A runs back and forh on an eas-wes secion of railroad rack. Train A s velociy, measured in meers per minue, is given by a differeniable funcion va( ), where ime is measured in minues. Seleced values for va( ) are given in he able above. (a) Find he average acceleraion of rain A over he inerval 8. ( minues ) 5 8 1 va( ) ( meers minue ) 1 4 1 15 (b) Do he daa in he able suppor he conclusion ha rain A s velociy is 1 meers per minue a some ime wih 5< < 8? Give a reason for your answer. (c) A ime =, rain A s posiion is meers eas of he Origin Saion, and he rain is moving o he eas. Wrie an expression involving an inegral ha gives he posiion of rain A, in meers from he Origin Saion, a ime = 1. Use a rapezoidal sum wih hree subinervals indicaed by he able o approximae he posiion of he rain a ime = 1. (d) A second rain, rain B, ravels norh from he Origin Saion. A ime he velociy of rain B is given by vb ( ) = 5 + 6 + 5, and a ime = he rain is 4 meers norh of he saion. Find he rae, in meers per minue, a which he disance beween rain A and rain B is changing a ime =. (a) average accel = v A ( 8) va( ) 1 1 11 m/min = = 8 6 1 : average acceleraion (b) v A is differeniable v A is coninuous v ( 8) = 1 < 1 < 4 = v ( 5) A Therefore, by he Inermediae Value Theorem, here is a ime, 5 < < 8, such ha v ( ) = 1. A A 1 : va( 8) < 1 < va( 5) : 1 : conclusion, using IVT 1 1 A A A A 1 va( ) d 4 (c) s ( 1) = s ( ) + v ( ) d = + v ( ) d 1 + 4 4 1 1 15 + + = 45 s ( 1) 45 = 15 A The posiion of Train A a ime = 1 minues is approximaely 15 meers wes of Origin Saion. (d) Le x be rain A s posiion, y rain B s posiion, and z he disance beween rain A and rain B. dz dx dy z = x + y z = x + y d d d x =, y = 4 z = 5 vb ( ) = + 1 + 5 = 15 dz 5 = ( )( 1) + ( 4)( 15) d dz 8 = = 16 meers per minue d 5 1 : posiion expression : 1 : rapezoidal sum 1 : posiion a ime = 1 : implici differeniaion of : disance relaionship 14 The College Board. Visi he College Board on he Web: www.collegeboard.org.
1 SCORING GUIDELINES Quesion (minues) C ( ) (ounces) 1 4 5 6 5. 8.8 11. 1.8 1.8 14.5 Ho waer is dripping hrough a coffeemaker, filling a large cup wih coffee. The amoun of coffee in he cup a ime, 6, is given by a differeniable funcion C, where is measured in minues. Seleced values of C ( ), measured in ounces, are given in he able above. (a) Use he daa in he able o approximae C (.5 ). Show he compuaions ha lead o your answer, and indicae unis of measure. (b) Is here a ime, 4, a which C ( ) =? Jusify your answer. (c) Use a midpoin sum wih hree subinervals of equal lengh indicaed by he daa in he able o approximae 1 6 1 6 he value of C( ) d. 6 Using correc unis, explain he meaning of ( ) 6 C d in he conex of he problem..4 (d) The amoun of coffee in he cup, in ounces, is modeled by B ( ) = 16 16 e. Using his model, find he rae a which he amoun of coffee in he cup is changing when = 5. C( 4) C( ) 1.8 11. (a) C (.5) = 4 1 = 1.6 ounces min : { 1 : approximaion 1 : unis (b) C is differeniable C is coninuous (on he closed inerval) C( 4) C( ) 1.8 8.8 = = 4 Therefore, by he Mean Value Theorem, here is a leas one ime, < < 4, for which C ( ) =. C( 4) C( ) 1 : : 4 1 : conclusion, using MVT (c) 1 6 1 ( ) [ ( 1) ( ) ( 5) ] 6 C d C + C + C 6 1 = ( 5. + 11. + 1. 8) 6 1 = ( 6.6 ) = 1.1 ou n ce s 6 1 6 ( ) 6 C d is he average amoun of coffee in he cup, in ounces, over he ime inerval 6 minues. (d) B ( ) = 16(.4) e = 6.4e.4.4.4( 5) 6.4 B ( 5) = 6.4 e = ounces min e : : 1 : midpoin sum 1 : approximaion 1 : inerpreaion 1 : B ( ) 1 : B ( 5) 1 The College Board. Visi he College Board on he Web: www.collegeboard.org.
1 SCORING GUIDELINES Quesion 1 (minues) 4 9 15 W() (degrees Fahrenhei) 55. 57.1 61.8 67.9 71. The emperaure of waer in a ub a ime is modeled by a sricly increasing, wice-differeniable funcion W, where W() is measured in degrees Fahrenhei and is measured in minues. A ime =, he emperaure of he waer is 55 F. The waer is heaed for minues, beginning a ime =. Values of W() a seleced imes for he firs minues are given in he able above. (a) Use he daa in he able o esimae W ( 1 ). Show he compuaions ha lead o your answer. Using correc unis, inerpre he meaning of your answer in he conex of his problem. (b) Use he daa in he able o evaluae W () d. Using correc unis, inerpre he meaning of () W d in he conex of his problem. 1 (c) For, he average emperaure of he waer in he ub is (). W d Use a lef Riemann sum 1 wih he four subinervals indicaed by he daa in he able o approximae (). W d Does his approximaion overesimae or underesimae he average emperaure of he waer over hese minues? Explain your reasoning. (d) For 5, he funcion W ha models he waer emperaure has firs derivaive given by W () =.4 cos(.6 ). Based on he model, wha is he emperaure of he waer a ime = 5? W( 15) W( 9) 67.9 61.8 (a) W ( 1) = 15 9 6 = 1.17 (or 1.16) The waer emperaure is increasing a a rae of approximaely 1.17 F per minue a ime = 1 minues. 1 : esimae : { 1 : inerpreaion wih unis (b) W () d = W( ) W( ) = 71. 55. = 16 The waer has warmed by 16 F over he inerval from = o = minues. (c) 1 () 1 ( 4 ( ) 5 ( 4 ) 6 ( 9 ) 5 ( 15 )) W d W + W + W + W 1 = ( 4 55. + 5 57.1 + 6 61.8 + 5 67.9 ) 1 = 115.8 = 6.79 This approximaion is an underesimae, because a lef Riemann sum is used and he funcion W is sricly increasing. (d) W ( 5) 71. W ( ) d = 71. +.4155 = 7.4 5 = + : { 1 : inegral : { 1 : value 1 : inerpreaion wih unis : 1 : lef Riemann sum 1 : approximaion 1 : underesimae wih reason 1 The College Board. Visi he College Board on he Web: www.collegeboard.org.
11 SCORING GUIDELINES (Form B) Quesion 5 (seconds) B( ) (meers) v ( ) (meers per second) 1 4 6 1 16 9 49...5 4.6 Ben rides a unicycle back and forh along a sraigh eas-wes rack. The wice-differeniable funcion B models Ben s posiion on he rack, measured in meers from he wesern end of he rack, a ime, measured in seconds from he sar of he ride. The able above gives values for B( ) and Ben s velociy, v ( ), measured in meers per second, a seleced imes. (a) Use he daa in he able o approximae Ben s acceleraion a ime = 5 seconds. Indicae unis of measure. 6 (b) Using correc unis, inerpre he meaning of v () d in he conex of his problem. Approximae 6 v () d using a lef Riemann sum wih he subinervals indicaed by he daa in he able. (c) For 4 6, mus here be a ime when Ben s velociy is meers per second? Jusify your answer. (d) A ligh is direcly above he wesern end of he rack. Ben rides so ha a ime, he disance L ( ) beween Ben and he ligh saisfies ( L ()) = 1 + ( B ()). A wha rae is he disance beween Ben and he ligh changing a ime = 4? v( 1) v( ). (a) a( 5) = =. meers sec 1 1 6 (b) v () d is he oal disance, in meers, ha Ben rides over he 6-second inerval = o = 6. 1 : meaning of inegral : { 1 : approximaion 6 v () d. 1 +.( 4 1) +.5( 6 4) = 19 meers B( 6) B( 4) 49 9 (c) Because = =, he Mean Value Theorem 6 4 implies here is a ime, 4 < < 6, such ha v ( ) =. (d) L () L () = B () B () B( 4) v( 4) 9.5 L ( 4 ) = = = meers sec L( 4) 144 + 81 1 : difference quoien : { 1 : conclusion wih jusificaion 1 : derivaives : 1 : uses B () = v() 1 : unis in (a) or (b) 11 The College Board. Visi he College Board on he Web: www.collegeboard.org.
11 SCORING GUIDELINES Quesion (minues) H () (degrees Celsius) 5 9 1 66 6 5 44 4 As a po of ea cools, he emperaure of he ea is modeled by a differeniable funcion H for 1, where ime is measured in minues and emperaure H ( ) is measured in degrees Celsius. Values of H () a seleced values of ime are shown in he able above. (a) Use he daa in he able o approximae he rae a which he emperaure of he ea is changing a ime =.5. Show he compuaions ha lead o your answer. 1 1 (b) Using correc unis, explain he meaning of () 1 H d in he conex of his problem. Use a rapezoidal 1 1 sum wih he four subinervals indicaed by he able o esimae (). 1 H d 1 (c) Evaluae H () d. Using correc unis, explain he meaning of he expression in he conex of his problem. (d) A ime =, biscuis wih emperaure 1 C were removed from an oven. The emperaure of he biscuis a ime is modeled by a differeniable funcion B for which i is known ha.17 B () = 1.84 e. Using he given models, a ime = 1, how much cooler are he biscuis han he ea? H( 5) H( ) (a) H (.5) 5 5 6 = =.666 or.667 degrees Celsius per minue 1 1 (b) () 1 H d is he average emperaure of he ea, in degrees Celsius, over he 1 minues. 1 1 1 66 6 6 5 5 44 44 4 1 H d 1 1 + + + + () ( + + 4 + 1 ) = 5.95 (c) H () d = H( 1) H( ) = 4 66 = The emperaure of he ea drops degrees Celsius from ime = o ime = 1 minues. 1 (d) B( 1) = 1 + B ( ) d = 4.1875; H( 1) B( 1) = 8.817 The biscuis are 8.817 degrees Celsius cooler han he ea. : 1 : meaning of expression 1 : rapezoidal sum 1 : esimae 1 : value of inegral : { 1 : meaning of expression 1 : inegrand : 1 : uses B( ) = 1 11 The College Board. Visi he College Board on he Web: www.collegeboard.org.
1 SCORING GUIDELINES (Form B) Quesion 4 6 8 1 1 P() 46 5 57 6 6 6 The figure above shows an aboveground swimming pool in he shape of a cylinder wih a radius of 1 fee and a heigh of 4 fee. The pool conains 1 cubic fee of waer a ime =. During he ime inerval 1 hours, waer is pumped ino he pool a he rae P () cubic fee per hour. The able above gives values of P () for seleced values of. During he same ime inerval, waer is leaking from he pool a he rae R() cubic fee.5 per hour, where R () = 5 e. (Noe: The volume V of a cylinder wih radius r and heigh h is given by. V = π r h ) (a) Use a midpoin Riemann sum wih hree subinervals of equal lengh o approximae he oal amoun of waer ha was pumped ino he pool during he ime inerval 1 hours. Show he compuaions ha lead o your answer. (b) Calculae he oal amoun of waer ha leaked ou of he pool during he ime inerval 1 hours. (c) Use he resuls from pars (a) and (b) o approximae he volume of waer in he pool a ime = 1 hours. Round your answer o he neares cubic foo. (d) Find he rae a which he volume of waer in he pool is increasing a ime = 8 hours. How fas is he waer level in he pool rising a = 8 hours? Indicae unis of measure in boh answers. 1 : { (a) P () d 46 4 + 57 4 + 6 4 = 66 f 1 : midpoin sum 1 : { 1 : inegral (b) R () d= 5.594 f 1 1 (c) 1 + P () d R () d= 144.46 A ime = 1 hours, he volume of waer in he pool is approximaely 144 f. (d) V () = P() R().4 V ( 8) = P( 8) R( 8) = 6 5e = 4.41 or 4.4 f hr V = π ( 1) h dv dh = 144π d d dh 1 dv.95 d = 144π d = or.96 f hr = 8 = 8 1 : V ( 8) dv 1 : equaion relaing and d 4 : dh 1 : d = 8 1 : unis of f hr and f hr dh d 1 The College Board. Visi he College Board on he Web: www.collegeboard.com.
1 SCORING GUIDELINES Quesion A zoo sponsored a one-day cones o name a new baby elephan. Zoo visiors deposied enries in a special box beween noon ( = ) and 8 P.M. ( = 8. ) The number of enries in he box hours afer noon is modeled by a differeniable funcion E for 8. Values of E(), in hundreds of enries, a various imes are shown in he able above. (a) Use he daa in he able o approximae he rae, in hundreds of enries per hour, a which enries were being deposied a ime = 6. Show he compuaions ha lead o your answer. 1 8 (b) Use a rapezoidal sum wih he four subinervals given by he able o approximae he value of (). 8 E d 1 8 Using correc unis, explain he meaning of () 8 E d in erms of he number of enries. (c) A 8 P.M., voluneers began o process he enries. They processed he enries a a rae modeled by he funcion P, where P () = + 98 976 hundreds of enries per hour for 8 1. According o he model, how many enries had no ye been processed by midnigh ( = 1 )? (d) According o he model from par (c), a wha ime were he enries being processed mos quickly? Jusify your answer. E( 7) E( 5) (a) E ( 6) = 4 hundred enries per hour 7 5 1 8 (b) () 8 E d 1 E( ) + E( ) E( ) + E( 5) E( 5) + E( 7) E( 7) + E( 8) 1 8 + + + = 1.687 or 1. 688 1 8 () 8 E d is he average number of hundreds of enries in he box beween noon and 8 P.M. (c) P () d= 16 = 7 (hours) E() (hundreds of enries) : 1 : rapezoidal sum 1 : approximaion 1 : meaning 1 hundred enries : 8 { 1 : inegral (d) P () = when = 9.185 and = 1.816497. P() 8 9.185 5.8866 1.816497.9118 1 8 Enries are being processed mos quickly a ime = 1. 5 7 8 4 1 1 : 1 : considers P () = 1 : idenifies candidaes wih jusificaion 1 The College Board. Visi he College Board on he Web: www.collegeboard.com.
9 SCORING GUIDELINES (Form B) Quesion 6 (seconds) v () (meers per second) 8 5 4 5 1 8 4 7 The velociy of a paricle moving along he x-axis is modeled by a differeniable funcion v, where he posiion x is measured in meers, and ime is measured in seconds. Seleced values of v () are given in he able above. The paricle is a posiion x = 7 meers when = seconds. (a) Esimae he acceleraion of he paricle a = 6 seconds. Show he compuaions ha lead o your answer. Indicae unis of measure. (b) Using correc unis, explain he meaning of v () din he conex of his problem. Use a 4 rapezoidal sum wih he hree subinervals indicaed by he daa in he able o approximae v () d. (c) For 4, mus he paricle change direcion in any of he subinervals indicaed by he daa in he able? If so, idenify he subinervals and explain your reasoning. If no, explain why no. (d) Suppose ha he acceleraion of he paricle is posiive for < < 8 seconds. Explain why he posiion of he paricle a = 8 seconds mus be greaer han x = meers. 4 1 : unis in (a) and (b) v( 4) v( ) 11 (a) a( 6) = v ( 6 ) = meers sec 4 8 4 (b) v () dis he paricle s change in posiion in meers from ime = seconds o ime = 4 seconds. 4 v () d v( ) + v( 5) v( 5) + v( ) v( ) + v( 4) 5 + 7 + 8 = 75 meers (c) v ( 8) > and v ( ) < v ( ) < and v ( 4) > Therefore, he paricle changes direcion in he inervals 8 < < and < < 4. (d) Since v () = a() > for < < 8, v () on his inerval. Therefore, x( 8) = x( ) + v( ) d 7 + 8 >. 8 : 1 : meaning of 4 : rapezoidal approximaion : { 1 : explanaion v () d 1 : v () = a() : 1 : explanaion of x( 8) > 9 The College Board. All righs reserved. Visi he College Board on he Web: www.collegeboard.com.
9 SCORING GUIDELINES Quesion 5 x 5 8 1 f ( x ) 1 4 6 Le f be a funcion ha is wice differeniable for all real numbers. The able above gives values of f for seleced poins in he closed inerval x 1. (a) Esimae f ( 4. ) Show he work ha leads o your answer. 1 (b) Evaluae ( 5 f ( x) ) dx. Show he work ha leads o your answer. (c) Use a lef Riemann sum wih subinervals indicaed by he daa in he able o approximae f ( x) dx. Show he work ha leads o your answer. (d) Suppose f ( 5) = and f ( x) < for all x in he closed inerval 5 x 8. Use he line angen o he graph of f a x = 5 o show ha f ( 7) 4. Use he secan line for he graph of f on 5 x 8 o 4 show ha f ( 7 ). f( 5) f( ) (a) f ( 4) = 5 1 1 1 (b) ( 5f ( x) ) dx = dx 5 f ( x) dx = ( 1 ) 5 ( f (1) f ()) = 8 1 : { (c) f( x) dx f( )( ) + f( )( 5 ) + f( 5)( 8 5) + f( 8)( 1 8) = 18 : 1 1 : uses Fundamenal Theorem of Calculus 1 : lef Riemann sum (d) An equaion for he angen line is y = + ( x 5 ). Since f ( x) < for all x in he inerval 5 x 8, he line angen o he graph of y = f( x) a x = 5 lies above he graph for all x in he inerval 5 < x 8. Therefore, f ( 7) + = 4. 5 An equaion for he secan line is y = + ( x 5 ). Since f ( x) < for all x in he inerval 5 x 8, he secan line connecing ( 5, f ( 5) ) and ( 8, f ( 8) ) lies below he graph of y = f( x) for all x in he inerval 5 < x < 8. 5 4 Therefore, f ( 7) + =. 1 : angen line 1 : shows f ( 7) 4 4 : 1 : secan line 4 1 : shows f ( 7) 9 The College Board. All righs reserved. Visi he College Board on he Web: www.collegeboard.com.
8 SCORING GUIDELINES (Form B) Disance from he river s edge (fee) Quesion 8 14 4 Deph of he waer (fee) 7 8 A scienis measures he deph of he Doe River a Picnic Poin. The river is 4 fee wide a his locaion. The measuremens are aken in a sraigh line perpendicular o he edge of he river. The daa are shown in he able above. The velociy of he waer a Picnic Poin, in fee per minue, is modeled by v = 16 + sin + 1 for 1 minues. () ( ) (a) Use a rapezoidal sum wih he four subinervals indicaed by he daa in he able o approximae he area of he cross secion of he river a Picnic Poin, in square fee. Show he compuaions ha lead o your answer. (b) The volumeric flow a a locaion along he river is he produc of he cross-secional area and he velociy of he waer a ha locaion. Use your approximaion from par (a) o esimae he average value of he volumeric flow a Picnic Poin, in cubic fee per minue, from = o = 1 minues. π x (c) The scienis proposes he funcion f, given by f( x) ( ) = 8sin, as a model for he deph of he 4 waer, in fee, a Picnic Poin x fee from he river s edge. Find he area of he cross secion of he river a Picnic Poin based on his model. (d) Recall ha he volumeric flow is he produc of he cross-secional area and he velociy of he waer a a locaion. To preven flooding, waer mus be divered if he average value of he volumeric flow a Picnic Poin exceeds 1 cubic fee per minue for a -minue period. Using your answer from par (c), find he average value of he volumeric flow during he ime inerval 4 6 minues. Does his value indicae ha he waer mus be divered? (a) ( + 7) ( 7 + 8) ( 8 + ) ( + ) 8 + 6 + 8 + = 115 f 1 : rapezoidal approximaion 1 1 (b) 115 () 1 v d = 187.169 or 187.17 f min π x 8sin dx = 1. or 1.1 f : { 1 : inegra1 4 (c) ( ) 4 (d) Le C be he cross-secional area approximaion from par (c). The average volumeric flow is 1 6 () 181.91 or 181.91 f min. C v d = 4 Yes, waer mus be divered since he average volumeric flow for his -minue period exceeds 1 f min. : : 1 : limis and average value consan 1 : inegrand 1 : volumeric flow inegral 1 : average volumeric flow wih reason 8 The College Board. All righs reserved. Visi he College Board on he Web: www.collegeboard.com.
8 SCORING GUIDELINES Quesion (hours) 1 4 7 8 9 L ()(people) 1 156 176 16 15 8 Concer ickes wen on sale a noon ( = ) and were sold ou wihin 9 hours. The number of people waiing in line o purchase ickes a ime is modeled by a wice-differeniable funcion L for 9. Values of L () a various imes are shown in he able above. (a) Use he daa in he able o esimae he rae a which he number of people waiing in line was changing a 5: P.M. ( = 5.5 ). Show he compuaions ha lead o your answer. Indicae unis of measure. (b) Use a rapezoidal sum wih hree subinervals o esimae he average number of people waiing in line during he firs 4 hours ha ickes were on sale. (c) For 9, wha is he fewes number of imes a which L () mus equal? Give a reason for your answer. (d) The rae a which ickes were sold for 9 is modeled by r () = 55e ickes per hour. Based on he model, how many ickes were sold by P.M. ( =, ) o he neares whole number? L( 7) L( 4) 15 16 (a) L ( 5.5) = = 8 people per hour 7 4 (b) The average number of people waiing in line during he firs 4 hours is approximaely 1 L( ) + L( 1 ) L() ( ) ( ) ( ) ( 1 ) 1 + L L ( 1) + L 4 ( 4 ) 4 + + = 155.5 people (c) L is differeniable on [, 9 ] so he Mean Value Theorem implies L () > for some in ( 1, ) and some in ( 4, 7 ). Similarly, L () < for some in (, 4 ) and some in ( 7, 8 ). Then, since L is coninuous on [, 9 ], he Inermediae Value Theorem implies ha L () = for a leas hree values of in [, 9 ]. OR The coninuiy of L on [ 1, 4 ] implies ha L aains a maximum value here. Since L( ) > L( 1) and L( ) > L( 4 ), his maximum occurs on ( 1, 4 ). Similarly, L aains a minimum on (, 7 ) and a maximum on ( 4, 8 ). L is differeniable, so L () = a each relaive exreme poin on (, 9 ). Therefore L () = for a leas hree values of in [, 9 ]. [Noe: There is a funcion L ha saisfies he given condiions wih L () = for exacly hree values of.] (d) r () d= 97.784 There were approximaely 97 ickes sold by P.M. : { 1 : esimae 1 : unis 1 : rapezoidal sum : { : : 1 : considers change in sign of L 1 : analysis 1 : conclusion OR 1 : considers relaive exrema of L on (, 9) 1 : analysis 1 : conclusion : { 1 : inegrand 1 : limis and answer 8 The College Board. All righs reserved. Visi he College Board on he Web: www.collegeboard.com.
7 SCORING GUIDELINES Quesion 5 (minues) r () (fee per minue) 5 7 11 1 5.7 4.. 1..6.5 The volume of a spherical ho air balloon expands as he air inside he balloon is heaed. The radius of he balloon, in fee, is modeled by a wice-differeniable funcion r of ime, where is measured in minues. For < < 1, he graph of r is concave down. The able above gives seleced values of he rae of change, r (), of he radius of he balloon over he ime inerval 1. The radius of he balloon is fee when 4 = 5. (Noe: The volume of a sphere of radius r is given by V = π r. ) (a) Esimae he radius of he balloon when = 5.4 using he angen line approximaion a = 5. Is your esimae greaer han or less han he rue value? Give a reason for your answer. (b) Find he rae of change of he volume of he balloon wih respec o ime when = 5. Indicae unis of measure. (c) Use a righ Riemann sum wih he five subinervals indicaed by he daa in he able o approximae 1 1 r () d. Using correc unis, explain he meaning of () r d in erms of he radius of he balloon. 1 (d) Is your approximaion in par (c) greaer han or less han r () d? Give a reason for your answer. (a) r( 5.4) r( 5) + r ( 5) Δ = + (.4) =.8 f : Since he graph of r is concave down on he inerval { 1 : esimae 1 : conclusion wih reason 5 < < 5.4, his esimae is greaer han r ( 5.4 ). dv d dv d 4 π r (b) = ( ) 1 = 5 dr d = 4π( ) = 7π f min (c) r ( ) d ( 4.) + (.) + ( 1.) + 4(.6) + 1(.5) = 19. f 1 r () d is he change in he radius, in fee, from = o = 1 minues. (d) Since r is concave down, r is decreasing on < < 1. Therefore, his approximaion, 19. f, is less han 1 r () d. : dv : d : { 1 : approximaion 1 : explanaion 1 : conclusion wih reason Unis of f min in par (b) and f in par (c) 1 : unis in (b) 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).
6 SCORING GUIDELINES (Form B) Quesion 6 (sec) v () ( f sec ) a () ( f sec ) 15 5 5 5 6 14 1 1 1 5 1 4 A car ravels on a sraigh rack. During he ime inerval 6 seconds, he car s velociy v, measured in fee per second, and acceleraion a, measured in fee per second per second, are coninuous funcions. The able above shows seleced values of hese funcions. 6 (a) Using appropriae unis, explain he meaning of v () din erms of he car s moion. Approximae 6 v () dusing a rapezoidal approximaion wih he hree subinervals deermined by he able. (b) Using appropriae unis, explain he meaning of a () din erms of he car s moion. Find he exac value of a () d. (c) For < < 6, mus here be a ime when v () = 5? Jusify your answer. (d) For < < 6, mus here be a ime when a () =? Jusify your answer. 6 (a) v () dis he disance in fee ha he car ravels from = sec o = 6 sec. Trapezoidal approximaion for 6 v () d: 1 1 1 A = ( 14 + 1) 5 + ( 1)( 15) + ( 1)( 1) = 185 f (b) a () dis he car s change in velociy in f/sec from = sec o = sec. a() d = v () d = v( ) v( ) = 14 ( ) = 6 f/sec (c) Yes. Since v( 5) = 1 < 5 < = v( 5 ), he IVT guaranees a in ( 5, 5 ) so ha v () = 5. (d) Yes. Since v( ) = v( 5 ), he MVT guaranees a in (, 5 ) so ha a () = v () =. Unis of f in (a) and f/sec in (b) : { 1 : explanaion 1 : value : { 1 : explanaion 1 : value 1 : v( 5) < 5 < v( 5) : 1 : Yes; refers o IVT or hypoheses 1 : v( ) = v( 5) : 1 : Yes; refers o MVT or hypoheses 1 : unis in (a) and (b) 6 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for AP sudens and parens). 7
6 SCORING GUIDELINES Quesion 4 (seconds) v () (fee per second) 1 4 5 6 7 8 5 14 9 5 4 44 47 49 Rocke A has posiive velociy v () afer being launched upward from an iniial heigh of fee a ime = seconds. The velociy of he rocke is recorded for seleced values of over he inerval 8 seconds, as shown in he able above. (a) Find he average acceleraion of rocke A over he ime inerval 8 seconds. Indicae unis of measure. 7 (b) Using correc unis, explain he meaning of v () d in erms of he rocke s fligh. Use a midpoin Riemann sum wih subinervals of equal lengh o approximae v () d. 1 (c) Rocke B is launched upward wih an acceleraion of a () = fee per second per second. A ime + 1 = seconds, he iniial heigh of he rocke is fee, and he iniial velociy is fee per second. Which of he wo rockes is raveling faser a ime = 8 seconds? Explain your answer. 7 1 (a) Average acceleraion of rocke A is v( 8) v( ) 49 5 11 f sec = = 8 8 (b) Since he velociy is posiive, v () drepresens he disance, in fee, raveled by rocke A from = 1 seconds o = 7 seconds. 7 1 1 : explanaion : 1 : uses v( ), v( 4 ), v( 6) 1 : value A midpoin Riemann sum is [ v( ) + v( 4) + v( 6) ] = [ + 5 + 44] = f (c) Le vb () be he velociy of rocke B a ime. vb () = d = 6 + 1 + C + 1 = vb ( ) = 6 + C vb () = 6 + 1 4 v ( 8) = 5 > 49 = v( 8) B 4 : 1 : 6 + 1 1 : consan of inegraion 1 : uses iniial condiion 1 : finds vb ( 8 ), compares o v( 8 ), and draws a conclusion Rocke B is raveling faser a ime = 8 seconds. Unis of f sec in (a) and f in (b) 1 : unis in (a) and (b) 6 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for AP sudens and parens). 5
5 SCORING GUIDELINES Quesion Disance x (cm) Temperaure T( x ) ( C) 1 5 6 8 1 9 7 6 55 A meal wire of lengh 8 cenimeers (cm) is heaed a one end. The able above gives seleced values of he emperaure T( x ), in degrees Celsius ( C, ) of he wire x cm from he heaed end. The funcion T is decreasing and wice differeniable. (a) Esimae T ( 7. ) Show he work ha leads o your answer. Indicae unis of measure. (b) Wrie an inegral expression in erms of T( x ) for he average emperaure of he wire. Esimae he average emperaure of he wire using a rapezoidal sum wih he four subinervals indicaed by he daa in he able. Indicae unis of measure. 8 8 (c) Find T ( x) dx, and indicae unis of measure. Explain he meaning of ( ) T x dx in erms of he emperaure of he wire. (d) Are he daa in he able consisen wih he asserion ha T ( x) > for every x in he inerval < x < 8? Explain your answer. (a) T( 8) T( 6) 55 6 7 = = Ccm 8 6 1 8 (b) ( ) 8 T x dx 8 Trapezoidal approximaion for T( x) dx: 1 + 9 9+ 7 7 + 6 6 + 55 A = 1+ 4 + 1+ 1 Average emperaure 75.6875 C 8 A = 8 (c) T ( x) dx = T( 8) T( ) = 55 1 = 45 C The emperaure drops 45 C from he heaed end of he wire o he oher end of he wire. 1, 5 is 7 9 = 5.75. 5 1 5, 6 is 6 7 = 8. 6 5 T c 1 = 5.75 for some c 1 in he inerval ( 1, 5 ) T c = 8 for some c in he inerval ( 5, 6 ). I follows ha c, c. Therefore T (d) Average rae of change of emperaure on [ ] Average rae of change of emperaure on [ ] No. By he MVT, ( ) and ( ) T mus decrease somewhere in he inerval ( 1 ) is no posiive for every x in [, 8 ]. : 8 1 1 : T( x) dx 8 1 : rapezoidal sum : { 1 : value 1 : meaning 1 : wo slopes of secan lines : { wih explanaion Unis of Ccmin (a), and C in (b) and (c) 1 : unis in (a), (b), and (c) 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). 4
4 SCORING GUIDELINES (Form B) Quesion A es plane flies in a sraigh line wih (min) 5 1 15 5 5 4 posiive velociy v (), in miles per v ()(mpm) 7. 9. 9.5 7. 4.5.4.4 4. 7. minue a ime minues, where v is a differeniable funcion of. Seleced values of v () for 4 are shown in he able above. (a) Use a midpoin Riemann sum wih four subinervals of equal lengh and values from he able o 4 approximae v () d. Show he compuaions ha lead o your answer. Using correc unis, 4 explain he meaning of v () din erms of he plane s fligh. (b) Based on he values in he able, wha is he smalles number of insances a which he acceleraion of he plane could equal zero on he open inerval < < 4? Jusify your answer. 7 (c) The funcion f, defined by f() = 6 + cos( ) + sin ( ), is used o model he velociy of he 1 4 plane, in miles per minue, for 4. According o his model, wha is he acceleraion of he plane a =? Indicaes unis of measure. (d) According o he model f, given in par (c), wha is he average velociy of he plane, in miles per minue, over he ime inerval 4? (a) Midpoin Riemann sum is 1 [ v( 5) + v( 15) + v( 5) + v( 5) ] = 1 [ 9. + 7. +.4 + 4.] = 9 The inegral gives he oal disance in miles ha he plane flies during he 4 minues. : 1 : v( 5) + v( 15) + v( 5) + v( 5) 1 : meaning wih unis (b) By he Mean Value Theorem, v () = somewhere in he inerval (, 15 ) and somewhere in he inerval ( 5, ). Therefore he acceleraion will equal for a leas wo values of. 1 : wo insances : 1 : jusificaion (c) f ( ) =.47 or.48 miles per minue wih unis 1 4 (d) Average velociy = () 4 f d = 5.916 miles per minue : 1 : limis 1 : inegrand 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). 4
SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 1 18 4 6 Diameer The able above gives he measuremens of he B(x) (mm) 4 8 6 4 6 diameer of he blood vessel a seleced poins along he lengh of he blood vessel, where x represens he disance from one end of he blood vessel and Bx () is a wice-differeniable funcion ha represens he diameer a ha poin. (a) Wrie an inegral expression in erms of Bx () ha represens he average radius, in mm, of he blood vessel beween x = and x = 6. (b) Approximae he value of your answer from par (a) using he daa from he able and a midpoin Riemann sum wih hree subinervals of equal lengh. Show he compuaions ha lead o your answer. 75 Bx () (c) Using correc unis, explain he meaning of dx 15 in erms of he blood vessel. (d) Explain why here mus be a leas one value x, for < x < 6, such ha B ( x) =. (a) 1 6 Bx () dx : 6 1 : limis and consan 1 : inegrand (b) 1 B(6) B(18) B() 1 + + = 6 1 [ 6( + + 4 )] = 14 6 : 1 : B(6) + B(18) + B() (c) Bx ( ) Bx ( ) is he radius, so is he area of he cross secion a x. The expression is he volume in mm of he blood vessel beween 15 : 1 : volume in mm 1 : beween x = 15 and x = 75 mm and 75 mm from he end of he vessel. (d) By he MVT, B ( c1) = for some c 1 in (6, 18) and B ( c) = for some c in (4, 6). The MVT applied o B ( x) shows ha B () x = for some x in he inerval ( c1 c ),. : : explains why here are wo values of x where B( x) has he same value 1 : explains why ha means B ( x) = for < x < 6 Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com. 4 Noe: max 1/ if only explains why B ( x) = a some x in (, 6).
SCORING GUIDELINES Quesion The rae of fuel consumpion, in gallons per minue, recorded during an airplane fligh is given by a wice-differeniable and sricly increasing funcion R of ime. The graph of R and a able of seleced values of R( ), for he ime inerval 9 minues, are shown above. (a) Use daa from he able o find an approximaion for R ( 45 ). Show he compuaions ha lead o your answer. Indicae unis of measure. (b) The rae of fuel consumpion is increasing fases a ime = 45 minues. Wha is he value of R ( 45 )? Explain your reasoning. (c) Approximae he value of 9 R () d using a lef Riemann sum wih he five subinervals indicaed by he daa in he able. Is his numerical approximaion less han he value of 9 R () d? Explain your reasoning. b (d) For < b 9 minues, explain he meaning of ( ) R d in erms of fuel consumpion for he 1 b plane. Explain he meaning of R ( ) d b in erms of fuel consumpion for he plane. Indicae unis of measure in boh answers. (a) R(5) R(4) 55 4 R(45) = 5 4 1 = 1.5 gal/min : (b) R (45) = since R () has a maximum a (c) = 45. 9 R ( ) d ()() + (1)() + (1)(4) + ()(55) + ()(65) = 7 Yes, his approximaion is less because he graph of R is increasing on he inerval. : : 1 : a difference quoien using numbers from able and inerval ha conains 45 1 : 1.5 gal/min 1 : R(45) = 1 : reason 1 : value of lef Riemann sum 1 : less wih reason (d) b R () d is he oal amoun of fuel in gallons consumed for he firs b minues. 1 b R () d b is he average value of he rae of fuel consumpion in gallons/min during he firs b minues. Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com. 4 : : meanings b 1 : meaning of R ( ) d 1 b 1 : meaning of R ( ) d b < 1 > if no reference o ime b 1 : unis in boh answers
AP CALCULUS AB 1 SCORING GUIDELINES Quesion The emperaure, in degrees Celsius ( C), of he waer in a pond is a differeniable funcion W of ime. The able above shows he waer (days) emperaure as recorded every days over a 15-day period. (a) Use daa from he able o find an approximaion for W = (1). Show he 6 9 compuaions ha lead o your answer. Indicae unis of measure. 1 (b) Approximae he average emperaure, in degrees Celsius, of he waer 15 over he ime inerval > > 15 days by using a rapezoidal approximaion wih subinervals of lengh days. ( /) (c) A suden proposes he funcion P, given by P ( ) 1e Г, as a model for he W () ( C) emperaure of he waer in he pond a ime, where is measured in days and P () is measured in degrees Celsius. Find P= (1). Using appropriae unis, explain he meaning of your answer in erms of waer emperaure. (d) Use he funcion P defined in par (c) o find he average value, in degrees Celsius, of P () over he ime inerval > > 15 days. 1 8 4 1 (a) Difference quoien; e.g. W(15) ГW(1) 1 W = (1) N Г 15 Г 1 C/day or W(1) ГW(9) W = (1) N Г 1 Г 9 C/day or : 1 : difference quoien (wih unis) W(15) ГW(9) 1 W = (1) N Г 15 Г 9 C/day (b) (1) (8) (4) () 1 76.5 1 Average emperaure N (76.5) 5.1 C 15 : 1 : rapezoidal mehod (c) 1 P= (1) 1e Г e Г/ Г/ 1 4 e Г Г Г.549 C/day : 1 : P = (1) (wih or wihou unis) 1 : inerpreaion This means ha he emperaure is decreasing a he rae of.549 C/day when = 1 days. (d) 1 15 15 Г / 1e d 5.757 C : 1 : inegrand 1 : limis and average value consan Copyrigh 1 by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.
AB{ / BC{ 1999. The rae a which waer ows ou of a pipe, in gallons per hour, is given by a diereniable funcion R of ime. The able above shows he rae as measured every hours for a 4{hour period. (a) Use a midpoin Riemann sum wih 4 subdivisions of equal lengh o approximae Z 4 R() d. Using correc unis, explain he meaning of your answer in erms of waer ow. (b) Is here some ime, <<4, such ha R () =? Jusify your answer. (c) The rae of waer ow R() can be approximaed by Q() = 1 ; 768 + ;. Use Q() oapproximae he 79 average rae of waer ow during he 4{hour ime period. Indicae unis of measure. R() (hours) (gallons per hour) 9.6 1.4 6 1.8 9 11. 1 11.4 15 11. 18 1.7 1 1. 4 9.6 (a) Z 4 R() d 6[R() + R(9) + R(15) + R(1)] =6[1:4+11:+11:+1:] = 58.6 gallons This is an approximaion o he oal ow in gallons of waer from he pipe in he 4{hour period. 8 1: R() + R(9) + R(15) + R(1) >< 1: answer >: 1: explanaion (b) Yes Since R() = R(4) = 9:6, he Mean Value Theorem guaranees ha here is a, <<4, such har () =. ( 1: answer 1: MVT or equivalen (c) Average rae of ow average value of Q() = 1 Z 4 1 4 79 (768 + ; ) d = 1:785 gal/hr or 1.784 gal/hr 8 1: limis and average value consan >< 1: Q() asinegrand >: 1: answer (unis) Gallons in par (a) and gallons/hr in par (c), or equivalen. 1: unis
Velociy (fee per second) v() 9 8 7 6 5 4 1 O 1998 Calculus AB Scoring Guidelines 5 1 15 5 5 4 45 5 Time (seconds) v() (seconds) (fee per second) 5 1 1 15 55 5 7 78 5 81 4 75 45 6 5 7. The graph of he velociy v(), in f/sec, of a car raveling on a sraigh road, for 5, is shown above. A able of values for v(), a 5 second inervals of ime, is shown o he righ of he graph. (a) During wha inervals of ime is he acceleraion of he car posiive? Give a reason for your answer. (b) Find he average acceleraion of he car, in f/sec, over he inerval 5. (c) Find one approximaion for he acceleraion of he car, in f/sec, a = 4. Show he compuaions you used o arrive a your answer. (d) Approximae 5 v() d wih a Riemann sum, using he midpoins of five subinervals of equal lengh. Using correc unis, explain he meaning of his inegral. (a) (b) (c) (d) Acceleraion is posiive on (, 5) and (45, 5) because he velociy v() is increasing on [, 5] and [45, 5] Avg. Acc. = v(5) v() 5 or 1.44 f/sec Difference quoien; e.g. v(45) v(4) 5 v(4) v(5) 5 = = v(45) v(5) = 1 or 6 75 5 75 81 5 6 81 1 Slope of angen line, e.g. 5 = 7 5 hrough (5, 9) and (4, 75): v() d = 7 5 = f/sec or = 6 5 f/sec or = 1 1 f/sec 9 75 = f/sec 5 4 1[v(5) + v(15) + v(5) + v(5) + v(45)] = 1(1 + + 7 + 81 + 6) = 5 fee This inegral is he oal disance raveled in fee over he ime o 5 seconds. 1: (, 5) 1: (45, 5) 1: reason Noe: ignore inclusion of endpoins 1: answer { 1: mehod 1: answer Noe: / if firs poin no earned 1: midpoin Riemann sum 1: answer 1: meaning of inegral Copyrigh 1998 College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.