1998 Calculus AB Scoring Guidelines

Size: px

Start display at page:

Download "1998 Calculus AB Scoring Guidelines"

Vivien Wilkerson
5 years ago
Views:

1 AB{ / BC{ 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 {hour period. (a) Use a midpoin Riemann sum wih subdivisions of equal lengh o approximae Z R() d. Using correc unis, explain he meaning of your answer in erms of waer ow. (b) Is here some ime, <<, such ha R () =? Jusify your answer. (c) The rae of waer ow R() can be approximaed by Q() = 1 ; 76 + ;. Use Q() oapproximae he 79 average rae of waer ow during he {hour ime period. Indicae unis of measure. R() (hours) (gallons per hour) (a) Z R() d 6[R() + R(9) + R(15) + R(1)] =6[1:+11:+11:+1:] = 5.6 gallons This is an approximaion o he oal ow in gallons of waer from he pipe in he {hour period. 1: R() + R(9) + R(15) + R(1) >< 1: answer >: 1: explanaion (b) Yes Since R() = R() = 9:6, he Mean Value Theorem guaranees ha here is a, <<, such har () =. ( 1: answer 1: MVT or equivalen (c) Average rae of ow average value of Q() = 1 Z 1 79 (76 + ; ) d = 1:75 gal/hr or 1.7 gal/hr 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 1

2 Velociy (fee per second) v() O 199 Calculus AB Scoring Guidelines Time (seconds) v() (seconds) (fee per second) 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 =. 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 (5, 5) because he velociy v() is increasing on [, 5] and [5, 5] v(5) v() Avg. Acc. = 5 or 1. f/sec Difference quoien; e.g. v(5) v() 5 v() v(5) 5 v(5) v(5) 1 or = = = Slope of angen line, e.g. 5 = 7 5 hrough (5, 9) and (, 75): v() d = 7 5 = f/sec or = 6 5 f/sec or = 1 1 f/sec 9 75 = f/sec 5 1[v(5) + v(15) + v(5) + v(5) + v(5)] = 1( ) = 5 fee This inegral is he oal disance raveled in fee over he ime o 5 seconds. 1: (, 5) 1: (5, 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 199 College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

3 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 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 1 : answer (wih unis) W(15) ГW(9) 1 W = (1) N Г 15 Г 9 C/day (b) (1) () () () Average emperaure N (76.5) 5.1 C 15 : 1 : rapezoidal mehod 1 : answer (c) 1 P= (1) 1e Г e Г/ Г/ e Г 1 Г Г.59 C/day : 1 : P = (1) (wih or wihou unis) 1 : inerpreaion This means ha he emperaure is decreasing a he rae of.59 C/day when = 1 days. (d) Г / 1e d C : 1 : inegrand 1 : limis and average value consan 1 : answer Copyrigh 1 by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

4 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) 6 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 in erms of he blood vessel. (d) Explain why here mus be a leas one value x, for < x < 6, such ha B ( x) =. 15 (a) 1 6 Bx () dx : 6 1 : limis and consan 1 : inegrand (b) 1 B(6) B(1) B() = 6 1 [ 6( + + )] = 1 6 : 1 : B(6) + B(1) + B() 1 : answer (c) Bx ( ) is he radius, so Bx ( ) 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, 1) and B ( c) = for some c in (, 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. Noe: max 1/ if only explains why B ( x) = a some x in (, 6).

5 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 ( 5 ). Show he compuaions ha lead o your answer. Indicae unis of measure. (b) The rae of fuel consumpion is increasing fases a ime = 5 minues. Wha is he value of R ( 5 )? 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() 55 R(5) = 5 1 = 1.5 gal/min : (b) R (5) = since R () has a maximum a (c) = 5. 9 R ( ) d ()() + (1)() + (1)() + ()(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 5 1 : 1.5 gal/min 1 : R(5) = 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. 5 Available a apcenral.collegeboard.com. : : meanings 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 b

6 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 are shown in he able above. (a) Use a midpoin Riemann sum wih four subinervals of equal lengh and values from he able o approximae v () d. Show he compuaions ha lead o your answer. Using correc unis, 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 < <? Jusify your answer. 7 (c) The funcion f, defined by f() = 6 + cos( ) + sin ( ), is used o model he velociy of he 1 plane, in miles per minue, for. 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? (a) Midpoin Riemann sum is 1 [ v( 5) + v( 15) + v( 5) + v( 5) ] = 1 [ ] = 9 The inegral gives he oal disance in miles ha he plane flies during he minues. : 1 : v( 5) + v( 15) + v( 5) + v( 5) 1 : answer 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 ( ) =.7 or. miles per minue 1 : answer wih unis 1 (d) Average velociy = () f d = miles per minue : 1 : limis 1 : inegrand 1 : answer Copyrigh by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 6

7 5 SCORING GUIDELINES Quesion Disance x (cm) Temperaure T( x ) ( C) A meal wire of lengh 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. (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 <? Explain your answer. (a) T( ) T( 6) = = Ccm 6 1 : answer 1 (b) ( ) T x dx Trapezoidal approximaion for T( x) dx: A = Average emperaure C A = (c) T ( x) dx = T( ) T( ) = 55 1 = 5 C The emperaure drops 5 C from he heaed end of he wire o he oher end of he wire. 1, 5 is 7 9 = , 6 is 6 7 =. 6 5 T c 1 = 5.75 for some c 1 in he inerval ( 1, 5 ) T c = 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 [, ]. : 1 1 : T( x) dx 1 : rapezoidal sum 1 : answer : { 1 : value 1 : meaning 1 : wo slopes of secan lines : { 1 : answer 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 7 (for AP sudens and parens).

8 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. (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. 6 (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 = ( 1 + 1) 5 + ( 1)( 15) + ( 1)( 1) = 15 f (b) a () dis he car s change in velociy in f/sec from = sec o = sec. a() d = v () d = v( ) v( ) = 1 ( ) = 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 (for AP sudens and parens). 7

9 6 SCORING GUIDELINES Quesion (seconds) v () (fee per second) 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 seconds, as shown in he able above. (a) Find he average acceleraion of rocke A over he ime inerval seconds. Indicae unis of measure. 7 (b) Using correc unis, explain he meaning of v () din 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 = seconds? Explain your answer. 7 1 (a) Average acceleraion of rocke A is 1 : answer v( ) v( ) f sec = = (b) Since he velociy is posiive, v () drepresens he disance, in fee, raveled by rocke A from = 1 seconds o = 7 seconds : explanaion : 1 : uses v( ), v( ), v( 6) 1 : value A midpoin Riemann sum is [ v( ) + v( ) + v( 6) ] = [ ] = f (c) Le vb () be he velociy of rocke B a ime. vb () = d = C + 1 = v ( ) = 6 + C B vb () = v ( ) = 5 > 9 = v( ) B : 1 : : consan of inegraion 1 : uses iniial condiion 1 : finds vb ( ), compares o v( ), and draws a conclusion Rocke B is raveling faser a ime = 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 (for AP sudens and parens). 5 9

10 7 SCORING GUIDELINES Quesion 5 (minues) r () (fee per minue) 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 = 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. 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 r () d. Using correc unis, explain he meaning of () r d in erms of he radius of he balloon. (d) Is your approximaion in par (c) greaer han or less han r () d? Give a reason for your answer. (a) r( 5.) r( 5) + r ( 5) Δ = + (.) =. f Since he graph of r is concave down on he inerval 5 < < 5., his esimae is greaer han r ( 5. ). 1 1 : { 1 : esimae 1 : conclusion wih reason dv d dv d π (b) = ( ) r 1 = 5 dr d = π( ) = 7π f min (c) r ( ) d (.) + (.) + ( 1.) + (.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 : answer : { 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 (for sudens and parens). 5

11 SCORING GUIDELINES (Form B) Disance from he river s edge (fee) Quesion 1 Deph of he waer (fee) 7 A scienis measures he deph of he Doe River a Picnic Poin. The river is 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) = sin ( ), as a model for he deph of he 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 6 minues. Does his value indicae ha he waer mus be divered? (a) ( + 7) ( 7 + ) ( + ) ( + ) = 115 f 1 : rapezoidal approximaion 1 1 (b) 115 () 1 v d = or f min π x sin dx = 1. or 1.1 f : { 1 : inegra1 (c) ( ) (d) Le C be he cross-secional area approximaion from par (c). The average volumeric flow is 1 6 () or f min. C v d = 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 : answer 1 : answer 1 : volumeric flow inegral 1 : average volumeric flow 1 : answer wih reason The College Board. All righs reserved. Visi he College Board on he Web: 51

12 SCORING GUIDELINES Quesion (hours) L ()(people) 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 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( ) (a) L ( 5.5) = = people per hour 7 (b) The average number of people waiing in line during he firs hours is approximaely 1 L( ) + L( 1 ) L() ( ) ( ) ( ) ( 1 ) 1 + L L ( 1) + L ( ) + + = people (c) L is differeniable on [, 9 ] so he Mean Value Theorem implies L () > for some in ( 1, ) and some in (, 7 ). Similarly, L () < for some in (, ) and some in ( 7, ). 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, ] implies ha L aains a maximum value here. Since L( ) > L( 1) and L( ) > L( ), his maximum occurs on ( 1, ). Similarly, L aains a minimum on (, 7 ) and a maximum on (, ). 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= 97.7 There were approximaely 97 ickes sold by P.M. : { 1 : esimae 1 : unis 1 : rapezoidal sum : { 1 : answer : : 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 The College Board. All righs reserved. Visi he College Board on he Web: 5

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr AP CALCULUS AB/CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 3 6 8 134 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured