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

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1 AP CALCULUS AB/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( 3) 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 1R( ) + R( 1) + 3R( 3) + R( 6) = 1( 134) + ( 119) + 3( 95) + ( 74) = 85 liers 8 1 : lef Riemann sum 3 : 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( ) : 1 : answer 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:

2 AP CALCULUS AB/CALCULUS BC 15 SCORING GUIDELINES Quesion 3 (minues) v ( ) (meers per minue) 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 3 B ( ) = 6 + 3, 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 (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) = = = 76 meers 1 : explanaion 3 : 1 : righ Riemann sum 1 : approximaion (c) Bob s acceleraion is B ( ) = 3 1. B ( 5) = 3( 5) 1( 5) = 15 meers/min (d) Avg ve ( 6 + 3) 15 The College Board. Visi he College Board on he Web: 1 : uses B ( ) : 1 : answer l = 1 d 1 : inegral : 1 : aniderivaive 1 3 = : answer 1 1 = + 3 = 35 meers/ min 1 4

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7 AP CALCULUS AB 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 3 fee when 4 3 = 5. (Noe: The volume of a sphere of radius r is given by V = π r. ) 3 (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 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( 5.4) r( 5) + r ( 5) = 3 + (.4) = 3.8 f Since he graph of r is concave down on he inerval 5 < < 5.4, his esimae is greaer han r ( 5.4 ). 1 1 : { 1 : esimae 1 : conclusion wih reason dv d dv d π r (b) = ( ) 1 = 5 dr d = 4π( 3) = 7π f 3 min (c) r () d 4. ( ) + 3. ( ) + 1. ( ) ( ) ( ) = 19.3 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, 19.3 f, is less han 1 r () d. 3 : dv : d 1 : answer : { 1 : approximaion 1 : explanaion 1 : conclusion wih reason Unis of 3 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

8 AP CALCULUS AB 6 SCORING GUIDELINES Quesion 4 (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 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 () din erms of he rocke s fligh. Use a midpoin Riemann sum wih 3 subinervals of equal lengh o approximae v () d. 1 3 (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 1 : answer v( 8) v( ) 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 : explanaion 3 : 1 : uses v( ), v( 4 ), v( 6) 1 : value A midpoin Riemann sum is [ v( ) + v( 4) + v( 6) ] = [ ] = f (c) Le vb () be he velociy of rocke B a ime. 3 vb () = d = C + 1 = v ( ) = 6 + C B vb () = v ( 8) = 5 > 49 = v( 8) B 4 : 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 (for AP sudens and parens). 5 49

9 AP CALCULUS AB 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 (b) Using appropriae unis, explain he meaning of a () din erms of he car s moion. Find he exac value 3 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 3 from = 3 sec o = 6 sec. Trapezoidal approximaion for v () 6 3 d: A = ( ) 5 + ( 1)( 15) + ( 1)( 1) = 185 f 3 (b) a () dis he car s change in velociy in f/sec from = sec o = 3 sec. 3 3 a () d= v () d= v( 3) v( ) = 14 ( ) = 6 f/sec (c) Yes. Since v( 35) = 1 < 5 < = v( 5 ), he IVT guaranees a in ( 35, 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( 35) < 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 48

10 AP CALCULUS AB 4 SCORING GUIDELINES (Form B) Quesion 3 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( ) + 3sin ( ), 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 = 3? 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( 35) ] = 1 [ ] = 9 The inegral gives he oal disance in miles ha he plane flies during he 4 minues. 3 : 1 : v( 5) + v( 15) + v( 5) + v( 35) 1 : answer 1 : meaning wih unis (b) By he Mean Value Theorem, v () = somewhere in he inerval (, 15 ) and somewhere in he inerval ( 5, 3 ). Therefore he acceleraion will equal for a leas wo values of. 1 : wo insances : 1 : jusificaion (c) f ( 3) =.47 or.48 miles per minue 1 : answer wih unis 1 4 (d) Average velociy = () 4 f d = miles per minue 3 : 1 : limis 1 : inegrand 1 : answer Copyrigh 4 by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 46 4

11 AP CALCULUS AB 3 SCORING GUIDELINES Quesion 3 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 Explain your reasoning. 9 R () d? 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) = = 1.5 gal/min : (b) R (45) = since R () has a maximum a (c) = R () d (3)() + (1)(3) + (1)(4) + ()(55) + ()(65) = 37 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 3 by College Enrance Examinaion Board. All righs reserved. 45 Available a apcenral.collegeboard.com. 4 3 : : 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

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13 Velociy (fee per second) v() O 1998 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 apple apple 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 apple apple 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 Z 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) Acceleraion is posiive on (, 35) and (45, 5) because he velociy v() is increasing on [, 35] and [45, 5] 8 >< 1: (, 35) 3 1: (45, 5) >: 1: reason Noe: ignore inclusion of endpoins (b) Avg. Acc. = v(5) v() 5 or 1.44 f/sec = 7 5 = 7 5 1: answer (c) (d) Di erence quoien; e.g. v(45) v(4) = 5 v(4) v(35) = 5 v(45) v(35) = 1 or Slope of angen line, e.g. Z 5 hrough (35, 9) and (4, 75): v() d = 3 f/sec or = 6 5 f/sec or = 1 1 f/sec = 3 f/sec 1[v(5) + v(15) + v(5) + v(35) + v(45)] = 1( ) = 53 fee This inegral is he oal disance raveled in fee over he ime o 5 seconds. ( 1: mehod 1: answer Noe: / if firs poin no earned 8 1: midpoin Riemann sum >< 3 1: answer >: 1: meaning of inegral Copyrigh 1998 College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks 4 of he College Enrance Examinaion Board.