AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1

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2 AP CALCULUS AB/CALCULUS BC 6 SCORING GUIDELINES Quesion (hours) R ( ) (liers / hour) Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for, 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. 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 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 hours. (d) For, 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( ) 95 9 { (a) R ( ) = = liers/hr : : esimae : unis (b) The oal amoun of waer removed is given by R( ) d. R ( ) d R( ) + R( ) + R( ) + R( 6) = ( 4) + ( 9) + ( 95) + ( 74) = 5 liers : lef Riemann sum : : esimae : overesimae wih reason This is an overesimae since R is a decreasing funcion. (c) Toal 5 W ( ) d 5 + { : inegral = liers : : esimae (d) W( ) R( ) >, W( ) R( ) <, and W ( ) R ( ) is coninuous. { : considers W ( ) R ( ) : wih explanaion Therefore, he Inermediae Value Theorem guaranees a leas one ime, < <, 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. 6 The College Board. Visi he College Board on he Web:

3 AP CALCULUS AB/CALCULUS BC 5 SCORING GUIDELINES Quesion (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 ( 6 ). (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, 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. 4 (a) v ( 6) = 5 meers/min : approximaion 4 4 (b) v( ) d is he oal disance Johanna jogs, in meers, over he ime inerval 4 minues. 4 v( ) d v( ) + v( ) + 4 v( 4) + 6 v( 4) = = = 76 meers : explanaion : : righ Riemann sum : approximaion (c) Bob s acceleraion is B ( ) =. B ( 5) = ( 5) ( 5) = 5 meers/min (d) Avg ve ( 6 + ) 5 The College Board. Visi he College Board on he Web: : uses B ( ) : l = d : inegral 4 : : aniderivaive = + 4 = + = 5 meers/ min 4

4 AP CALCULUS AB/CALCULUS BC 4 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. ( minues ) 5 va( ) ( meers minue ) 4 5 (b) Do he daa in he able suppor he conclusion ha rain A s velociy is meers per minue a some ime wih 5< <? 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 =. Use a rapezoidal sum wih hree subinervals indicaed by he able o approximae he posiion of he rain a ime =. (d) A second rain, rain B, ravels norh from he Origin Saion. A ime he velociy of rain B is given by vb ( ) = , 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 ( ) va( ) m/min = = 6 : average acceleraion (b) v A is differeniable v A is coninuous v ( ) = < < 4 = v ( 5) A Therefore, by he Inermediae Value Theorem, here is a ime, 5 < <, such ha v ( ) =. A A : va( ) < < va( 5) : : conclusion, using IVT A A A A va( ) d 4 (c) s ( ) = s ( ) + v ( ) d = + v ( ) d = 45 s ( ) 45 = 5 A The posiion of Train A a ime = minues is approximaely 5 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 ( ) = = 5 dz 5 = ( )( ) + ( 4)( 5) d dz = = 6 meers per minue d 5 : posiion expression : : rapezoidal sum : posiion a ime = : implici differeniaion of : disance relaionship 4 The College Board. Visi he College Board on he Web:

5 SCORING GUIDELINES Quesion (minues) C ( ) (ounces) 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 6 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 ( ) = 6 6 e. Using his model, find he rae a which he amoun of coffee in he cup is changing when = 5. C( 4) C( ).. (a) C (.5) = 4 =.6 ounces min : { : approximaion : unis (b) C is differeniable C is coninuous (on he closed inerval) C( 4) C( ).. = = 4 Therefore, by he Mean Value Theorem, here is a leas one ime, < < 4, for which C ( ) =. C( 4) C( ) : : 4 : conclusion, using MVT (c) 6 ( ) [ ( ) ( ) ( 5) ] 6 C d C + C + C 6 = ( ) 6 = ( 6.6 ) =. ou n ce s 6 6 ( ) 6 C d is he average amoun of coffee in he cup, in ounces, over he ime inerval 6 minues. (d) B ( ) = 6(.4) e = 6.4e.4.4.4( 5) 6.4 B ( 5) = 6.4 e = ounces min e : : : midpoin sum : approximaion : inerpreaion : B ( ) : B ( 5) The College Board. Visi he College Board on he Web:

6 SCORING GUIDELINES Quesion 4 x....4 ( x) f 4.5 The funcion f is wice differeniable for x > wih f () = 5 and f () =. Values of f, he derivaive of f, are given for seleced values of x in he able above. (a) Wrie an equaion for he line angen o he graph of f a x =. Use his line o approximae f (.4 ). (b) Use a midpoin Riemann sum wih wo subinervals of equal lengh and values from he able o.4.4 approximae f ( x) dx. Use he approximaion for ( ) f x dx o esimae he value of f (.4 ). Show he compuaions ha lead o your answer. (c) Use Euler s mehod, saring a x = wih wo seps of equal size, o approximae f (.4 ). Show he compuaions ha lead o your answer. (d) Wrie he second-degree Taylor polynomial for f abou x =. Use he Taylor polynomial o approximae f (.4 ). (a) f () = 5, f () = An equaion for he angen line is y = 5 + ( x ). f (.4) 5 + (.4 ) =. : angen line : { : approximaion.4 (b) f ( x) dx (.)( ) + (.)( ) = f(.4) = f( ) + f ( x) dx : : midpoin Riemann sum : Fundamenal Theorem of Calculus f (.4 ) = 9. 6 (c) f(.) f( ) + (.)( ) = 6.6 f (.4) (.)( ) = 9. : Euler s mehod wih wo seps : { (d) T ( x) = 5 + ( x ) + ( x )! = 5 + ( x ) + ( x ) : Taylor polynomial : { : approximaion f ( 4. ) 5 + (.4 ) + (.4 ) = 9. The College Board. Visi he College Board on he Web:

7 SCORING GUIDELINES Quesion (minues) W() (degrees Fahrenhei) 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 ( ). 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. (c) For, he average emperaure of he waer in he ub is (). W d Use a lef Riemann sum 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( 5) W( 9) (a) W ( ) = =.7 (or.6) The waer emperaure is increasing a a rae of approximaely.7 F per minue a ime = minues. : esimae : { : inerpreaion wih unis (b) W () d = W( ) W( ) = = 6 The waer has warmed by 6 F over he inerval from = o = minues. (c) () ( 4 ( ) 5 ( 4 ) 6 ( 9 ) 5 ( 5 )) W d W + W + W + W = ( ) = 5. = 6.79 This approximaion is an underesimae, because a lef Riemann sum is used and he funcion W is sricly increasing. (d) W ( 5) 7. W ( ) d = = = + : { : inegral : { : value : inerpreaion wih unis : : lef Riemann sum : approximaion : underesimae wih reason The College Board. Visi he College Board on he Web:

8 SCORING GUIDELINES (Form B) Quesion 5 (seconds) B( ) (meers) v ( ) (meers per second) 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 ()) = + ( B ()). A wha rae is he disance beween Ben and he ligh changing a ime = 4? v( ) v( ). (a) a( 5) = =. meers sec 6 (b) v () d is he oal disance, in meers, Ben rides over he 6-second inerval = o = 6. : meaning of inegral : { : approximaion 6 v () d. +.( 4 ) +.5( 6 4) = 9 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) 44 + : difference quoien : { : conclusion wih jusificaion : derivaives : : uses B () = v() : unis in (a) or (b) The College Board. Visi he College Board on he Web:

9 SCORING GUIDELINES Quesion (minues) H () (degrees Celsius) As a po of ea cools, he emperaure of he ea is modeled by a differeniable funcion H for, 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. (b) Using correc unis, explain he meaning of () H d in he conex of his problem. Use a rapezoidal sum wih he four subinervals indicaed by he able o esimae (). H d (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 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.7 B () =.4 e. Using he given models, a ime =, how much cooler are he biscuis han he ea? H( 5) H( ) (a) H (.5) = =.666 or.667 degrees Celsius per minue (b) () H d is he average emperaure of he ea, in degrees Celsius, over he minues H d () ( ) = 5.95 (c) H () d = H( ) H( ) = 4 66 = The emperaure of he ea drops degrees Celsius from ime = o ime = minues. (d) B( ) = + B ( ) d = 4.75; H( ) B( ) =.7 The biscuis are.7 degrees Celsius cooler han he ea. : : meaning of expression : rapezoidal sum : esimae : value of inegral : { : meaning of expression : inegrand : : uses B( ) = The College Board. Visi he College Board on he Web:

10 SCORING GUIDELINES (Form B) Quesion 4 6 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 = hours. How fas is he waer level in he pool rising a = 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 ( ) = P( ) R( ) = 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 = = : V ( ) dv : equaion relaing and d 4 : dh : d = : unis of f hr and f hr dh d The College Board. Visi he College Board on he Web:

11 SCORING GUIDELINES (hours) E() (hundreds of enries) 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 P.M. ( =. ) The number of enries in he box hours afer noon is modeled by a differeniable funcion E for. 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. (b) Use a rapezoidal sum wih he four subinervals given by he able o approximae he value of (). E d Using correc unis, explain he meaning of () E d in erms of he number of enries. (c) A P.M., voluneers began o process he enries. They processed he enries a a rae modeled by he funcion P, where P () = hundreds of enries per hour for. According o he model, how many enries had no ye been processed by midnigh ( = )? (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 (b) () E d E( ) + E( ) E( ) + E( 5) E( 5) + E( 7) E( 7) + E( ) =.67 or. 6 () E d is he average number of hundreds of enries in he box beween noon and P.M. (c) P () d= 6 = 7 : : rapezoidal sum : approximaion : meaning hundred enries : { : inegral (d) P () = when = 9.5 and = P() Enries are being processed mos quickly a ime =. : : considers P () = : idenifies candidaes wih jusificaion The College Board. Visi he College Board on he Web:

12 9 SCORING GUIDELINES Quesion 5 x 5 f ( x ) 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. (a) Esimae f ( 4. ) Show he work ha leads o your answer. (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. 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 o 4 show ha f ( 7 ). f( 5) f( ) (a) f ( 4) = 5 (b) ( 5f ( x) ) dx = dx 5 f ( x) dx = ( ) 5 ( f () f ()) = : { (c) f( x) dx f( )( ) + f( )( 5 ) + f( 5)( 5) + f( )( ) = : : uses Fundamenal Theorem of Calculus : 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, 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. 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, he secan line connecing ( 5, f ( 5) ) and (, f ( ) ) lies below he graph of y = f( x) for all x in he inerval 5 < x <. 5 4 Therefore, f ( 7) + =. : angen line : shows f ( 7) 4 4 : : secan line 4 : shows f ( 7) 9 The College Board. All righs reserved. Visi he College Board on he Web:

13 SCORING GUIDELINES (Form B) Disance from he river s edge (fee) Quesion 4 4 Deph of he waer (fee) 7 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) ( ) = sin, 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 + ) ( + ) ( + ) = 5 f : rapezoidal approximaion (b) 5 () v d = 7.69 or 7.7 f min π x sin 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 ().9 or.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 The College Board. All righs reserved. Visi he College Board on he Web:

14 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 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) 5 6 (a) L ( 5.5) = = people per hour 7 4 (b) The average number of people waiing in line during he firs 4 hours is approximaely L( ) + L( ) L() ( ) ( ) ( ) ( ) + L L ( ) + L 4 ( 4 ) = 55.5 people (c) L is differeniable on [, 9 ] so he Mean Value Theorem implies L () > for some in (, ) and some in ( 4, 7 ). Similarly, L () < for some in (, 4 ) 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 [, 4 ] implies ha L aains a maximum value here. Since L( ) > L( ) and L( ) > L( 4 ), his maximum occurs on (, 4 ). Similarly, L aains a minimum on (, 7 ) and a maximum on ( 4, ). 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= There were approximaely 97 ickes sold by P.M. : { : esimae : unis : rapezoidal sum : { : : : considers change in sign of L : analysis : conclusion OR : considers relaive exrema of L on (, 9) : analysis : conclusion : { : inegrand : limis and answer The College Board. All righs reserved. Visi he College Board on he Web:

15 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 < <, 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. 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 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) Δ = + (.4) =. f : Since he graph of r is concave down on he inerval { : esimae : conclusion wih reason 5 < < 5.4, his esimae is greaer han r ( 5.4 ). dv d dv d 4 π r (b) = ( ) = 5 dr d = 4π( ) = 7π f min (c) r ( ) d ( 4.) + (.) + (.) + 4(.6) + (.5) = 9. f r () d is he change in he radius, in fee, from = o = minues. (d) Since r is concave down, r is decreasing on < <. Therefore, his approximaion, 9. f, is less han r () d. : dv : d : { : approximaion : explanaion : conclusion wih reason Unis of f min in par (b) and f in par (c) : unis in (b) and (c) 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

16 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 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 () d in erms of he rocke s fligh. Use a midpoin Riemann sum wih subinervals of equal lengh o approximae v () d. (c) Rocke B is launched upward wih an acceleraion of a () = fee per second per second. A ime + = 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 (a) Average acceleraion of rocke A is v( ) v( ) 49 5 f sec = = (b) Since he velociy is posiive, v () drepresens he disance, in fee, raveled by rocke A from = seconds o = 7 seconds. 7 : explanaion : : uses v( ), v( 4 ), v( 6) : value A midpoin Riemann sum is [ v( ) + v( 4) + v( 6) ] = [ ] = f (c) Le vb () be he velociy of rocke B a ime. vb () = d = C + = vb ( ) = 6 + C vb () = v ( ) = 5 > 49 = v( ) B 4 : : 6 + : consan of inegraion : uses iniial condiion : 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) : 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

17 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 (b) ( ) T x dx Trapezoidal approximaion for T( x) dx: A = Average emperaure C A = (c) T ( x) dx = T( ) T( ) = 55 = 45 C The emperaure drops 45 C from he heaed end of he wire o he oher end of he wire., 5 is 7 9 = , 6 is 6 7 =. 6 5 T c = 5.75 for some c in he inerval (, 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 ( ) is no posiive for every x in [, ]. : : T( x) dx : rapezoidal sum : { : value : meaning : wo slopes of secan lines : { wih explanaion Unis of Ccmin (a), and C in (b) and (c) : unis in (a), (b), and (c) Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 4

18 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.4 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

19 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() B() + + = 6 [ 6( )] = 4 6 : : B(6) + B() + 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, ) 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).

20 AP CALCULUS BC 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 5-day period. (a) Use daa from he able o find an approximaion for W = (). Show he 6 9 compuaions ha lead o your answer. Indicae unis of measure. (b) Approximae he average emperaure, in degrees Celsius, of he waer 5 over he ime inerval > > 5 days by using a rapezoidal approximaion wih subinervals of lengh days. ( /) (c) A suden proposes he funcion P, given by P ( ) e 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= (). 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 > > 5 days. 4 (a) Difference quoien; e.g. W(5) ГW() W = () N Г 5 Г C/day or W() ГW(9) W = () N Г Г 9 C/day or : : difference quoien (wih unis) W(5) ГW(9) W = () N Г 5 Г 9 C/day (b) () () (4) () 76.5 Average emperaure N (76.5) 5. C 5 : : rapezoidal mehod (c) P= () e Г e Г/ Г/ 4 e Г Г Г.549 C/day : : P = () (wih or wihou unis) : inerpreaion This means ha he emperaure is decreasing a he rae of.549 C/day when = days. (d) 5 5 Г / e d C : : inegrand : limis and average value consan Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

21 AB{ / BC{ 999. 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() = ; 76 + ;. 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) (a) Z 4 R() d 6[R() + R(9) + R(5) + R()] =6[:4+:+:+:] = 5.6 gallons This is an approximaion o he oal ow in gallons of waer from he pipe in he 4{hour period. : R() + R(9) + R(5) + R() >< : answer >: : explanaion (b) Yes Since R() = R(4) = 9:6, he Mean Value Theorem guaranees ha here is a, <<4, such har () =. ( : answer : MVT or equivalen (c) Average rae of ow average value of Q() = Z (76 + ; ) d = :75 gal/hr or.74 gal/hr : limis and average value consan >< : Q() asinegrand >: : answer (unis) Gallons in par (a) and gallons/hr in par (c), or equivalen. : unis