AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

Transcription

1 SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) Diameer The able above gives he measuremens of he B(x) (mm) 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 5 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) 6 Bx () dx 6 : limis and consan : inegrand (b) B(6) B(8) B() + + = 6 [ 6( )] = 4 6 : B(6) + B(8) + 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 5 : volume in mm : beween x = 5 and x = 75 mm and 75 mm from he end of he vessel. (d) By he MVT, B ( c) = for some c in (6, 8) 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 ( c c ),. : explains why here are wo values of x where B( x) has he same value : 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 / if only explains why B ( x) = a some x in (, 6).

2 4 SCORING GUIDELINES (Form B) Quesion A es plane flies in a sraigh line wih (min) posiive velociy v (), in miles per v ()(mpm) 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 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 [ v( 5) + v( 5) + v( 5) + v( 5) ] = [ ] = 9 The inegral gives he oal disance in miles ha he plane flies during he 4 minues. : : v( 5) + v( 5) + v( 5) + v( 5) : meaning wih unis (b) By he Mean Value Theorem, v () = somewhere in he inerval (, 5 ) and somewhere in he inerval ( 5, ). Therefore he acceleraion will equal for a leas wo values of. : wo insances : jusificaion (c) f ( ) =.47 or.48 miles per minue wih unis 4 (d) Average velociy = () 4 f d = 5.96 miles per minue : : limis : inegrand Copyrigh 4 by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 4

3 6 SCORING GUIDELINES (Form B) Quesion 6 (sec) v () ( f sec ) a () ( f sec ) 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: A = ( 4 + ) 5 + ( )( 5) + ( )( ) = 85 f (b) a () dis he car s change in velociy in f/sec from = sec o = sec. a() d = v () d = v( ) v( ) = 4 ( ) = 6 f/sec (c) Yes. Since v( 5) = < 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) { : explanaion : value { : explanaion : value : v( 5) < 5 < v( 5) : Yes; refers o IVT or hypoheses : v( ) = v( 5) : Yes; refers o MVT or hypoheses : unis in (a) and (b) 6 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 7

4 8 SCORING GUIDELINES (Form B) Disance from he river s edge (fee) Quesion 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 = 6 + sin + for 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 = 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 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 + ) ( + ) = 5 f : rapezoidal approximaion (b) 5 () v d = or 87.7 f min π x 8sin dx =. or. f { : inegra 4 (c) ( ) 4 (d) Le C be he cross-secional area approximaion from par (c). The average volumeric flow is 6 () 8.9 or 8.9 f min. C v d = 4 Yes, waer mus be divered since he average volumeric flow for his -minue period exceeds f min. : : : limis and average value consan : inegrand : volumeric flow inegral : average volumeric flow wih reason 8 The College Board. All righs reserved. Visi he College Board on he Web:

5 9 SCORING GUIDELINES (Form B) Quesion 6 (seconds) v () (meers per second) 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 : unis in (a) and (b) v( 4) v( ) (a) a( 6) = v ( 6 ) = meers sec (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) = 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 >. 8 : : meaning of 4 rapezoidal approximaion { : explanaion v () d : v () = a() : explanaion of x( 8) > 9 The College Board. All righs reserved. Visi he College Board on he Web:

6 SCORING GUIDELINES (Form B) Quesion P() The figure above shows an aboveground swimming pool in he shape of a cylinder wih a radius of fee and a heigh of 4 fee. The pool conains cubic fee of waer a ime =. During he ime inerval 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 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 hours. (c) Use he resuls from pars (a) and (b) o approximae he volume of waer in he pool a ime = 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. { (a) P () d = 66 f : midpoin sum { : inegral (b) R () d= f (c) + P () d R () d= A ime = hours, he volume of waer in he pool is approximaely 44 f. (d) V () = P() R().4 V ( 8) = P( 8) R( 8) = 6 5e = 4.4 or 4.4 f hr V = π ( ) h dv dh = 44π d d dh dv.95 d = 44π d = or.96 f hr = 8 = 8 : V ( 8) dv : equaion relaing and d 4 : dh : d = 8 : unis of f hr and f hr dh d The College Board. Visi he College Board on he Web:

7 SCORING GUIDELINES (Form B) Quesion 4 Consider a differeniable funcion f having domain all posiive real numbers, and for which i is known ha f ( x) = ( 4 x) x for x >. (a) Find he x-coordinae of he criical poin of f. Deermine wheher he poin is a relaive maximum, a relaive minimum, or neiher for he funcion f. Jusify your answer. (b) Find all inervals on which he graph of f is concave down. Jusify your answer. (c) Given ha f ( ) =, deermine he funcion f. (a) f ( x) = a x = 4 f ( x) > for < x < 4 f ( x) < for x > 4 Therefore f has a relaive maximum a x = 4. : : x = 4 : relaive maximum : jusificaion 4 (b) f ( x) = x + ( 4 x)( x ) 4 = x x + x 4 = x ( x 6) ( x 6) = 4 x f ( x) < for < x < 6 f ( x) : wih jusificaion The graph of f is concave down on he inerval < x < 6. x (c) f( x) = + ( 4 ) = + + = x + x d = x = : : inegral : aniderivaive The College Board. Visi he College Board on he Web: