Calculus BC 2015 Scoring Guidelines

Transcription

1 AP Calculus BC 5 Scorig Guidelies 5 The College Board. College Board, Advaced Placeme Program, AP, AP Ceral, ad he acor logo are regisered rademarks of he College Board. AP Ceral is he official olie home for he AP Program: apceral.collegeboard.org.

2 AP CALCULUS AB/CALCULUS BC 5 SCORING GUIDELINES Quesio The rae a which raiwaer flows io a draipipe is modeled by he fucio R, where R = si 35 cubic fee per hour, is measured i hours, ad 8. The pipe is parially blocked, allowig waer o drai ou he 3 oher ed of he pipe a a rae modeled by D = cubic fee per hour, for 8. There are 3 cubic fee of waer i he pipe a ime =. (a) How may cubic fee of raiwaer flow io he pipe durig he 8-hour ime ierval 8? (b) Is he amou of waer i he pipe icreasig or decreasig a ime = 3 hours? Give a reaso for your aswer. (c) A wha ime, 8, is he amou of waer i he pipe a a miimum? Jusify your aswer. (d) The pipe ca hold 5 cubic fee of waer before overflowig. For > 8, waer coiues o flow io ad ou of he pipe a he give raes uil he pipe begis o overflow. Wrie, bu do o solve, a equaio ivolvig oe or more iegrals ha gives he ime w whe he pipe will begi o overflow. 8 : : iegrad (a) R d = : aswer (b) R( 3) D( 3) =.3363 < Sice R( 3) < D( 3, ) he amou of waer i he pipe is decreasig a ime = 3 hours. (c) The amou of waer i he pipe a ime, 8, is 3 + [ R D ] d. : cosiders R( 3 ) ad D( 3) : : aswer ad reaso : cosiders R D = 3 : : aswer : jusificaio R D = =, Amou of waer i he pipe The amou of waer i he pipe is a miimum a ime = 3.7 (or 3.7) hours. w (d) 3 + [ R ( ) D ( )] d = 5 : : iegral : equaio 5 The College Board.

3 AP CALCULUS BC 5 SCORING GUIDELINES Quesio A ime, a paricle movig alog a curve i he y-plae has posiio (, y ) wih velociy vecor.5 ( ) v = cos, e. A =, he paricle is a he poi ( 3, 5 ). (a) Fid he -coordiae of he posiio of he paricle a ime =. (b) For < <, here is a poi o he curve a which he lie age o he curve has a slope of. A wha ime is he objec a ha poi? (c) Fid he ime a which he speed of he paricle is 3. (d) Fid he oal disace raveled by he paricle from ime = o ime =. (a) = 3 + cos d =.557 (or.556) : iegral 3 : : uses iiial codiio : aswer (b).5 dy dy d e = = d d d cos e cos.5 ( ) = : slope i erms of : : aswer =.84 (c) Speed cos ( ) = + e ( ) cos + e = 3 =.96 (or.95) : speed i erms of : : aswer (d) Disace = cos ( ) + e d =.595 (or.594) : : iegral : aswer 5 The College Board.

4 AP CALCULUS AB/CALCULUS BC 5 SCORING GUIDELINES Quesio 3 (miues) v (meers per miue) Johaa jogs alog a sraigh pah. For 4, Johaa s velociy is give by a differeiable fucio v. Seleced values of v, where is measured i miues ad v is measured i meers per miue, are give i he able above. (a) Use he daa i he able o esimae he value of v ( 6 ). (b) Usig correc uis, eplai he meaig of he defiie iegral v d i he coe of he problem. 4 Approimae he value of v d usig a righ Riema sum wih he four subiervals idicaed i he able. (c) Bob is ridig his bicycle alog he same pah. For, Bob s velociy is modeled by 3 B = 6 + 3, where is measured i miues ad B is measured i meers per miue. Fid Bob s acceleraio a ime = 5. (d) Based o he model B from par (c), fid Bob s average velociy durig he ierval. 4 (a) v ( 6) = 5 meers/mi : approimaio 4 4 (b) v d is he oal disace Johaa jogs, i meers, over he ime ierval 4 miues. 4 v d v + 8 v + 4 v( 4) + 6 v( 4) = = = 76 meers : eplaaio 3 : : righ Riema sum : approimaio (c) Bob s acceleraio is B = 3. B ( 5) = 3( 5) ( 5) = 5 meers/mi 3 (d) Avg ve ( 6 + 3) 5 The College Board. : uses B : : aswer l = d : iegral 4 3 : : aiderivaive 3 = : aswer = + 3 = 35 meers/ mi 4

5 dy Cosider he differeial equaio y. d = AP CALUCLUS AB/CALCULUS BC 5 SCORING GUIDELINES Quesio 4 (a) O he aes provided, skech a slope field for he give differeial equaio a he si pois idicaed. d y (b) Fid i erms of ad y. Deermie he cocaviy of all soluio curves for he give differeial d equaio i Quadra II. Give a reaso for your aswer. (c) Le y = f be he paricular soluio o he differeial equaio wih he iiial codiio f = 3. Does f have a relaive miimum, a relaive maimum, or eiher a =? Jusify your aswer. (d) Fid he values of he cosas m ad b for which y = m + b is a soluio o he differeial equaio. (a) : : slopes where = : slopes whe re = (b) (c) d y dy = = ( y) = + y d d I Quadra II, < ad y >, so + y >. Therefore, all soluio curves are cocave up i Quadra II. dy d = = 3= =/ (, y) (, 3) Therefore, f has eiher a relaive miimum or a relaive maimum a =. d y : : d : cocave up wih reaso : dy : cosiders d (, y) = (, 3) : coclusio wih jusificaio (d) y = m + b dy d = ( m + b ) = m d d y = m ( m + b) = m ( m) ( m + b) = m = m = b = m b = d : ( m + b ) = m d 3: : y = m : aswer Therefore, m = ad b =. 5 The College Board.

6 AP CALCULUS BC 5 SCORING GUIDELINES Quesio 5 Cosider he fucio f =, where k is a ozero cosa. The derivaive of f is give by k f = k ( k). (a) Le k = 3, so ha f =. Wrie a equaio for he lie age o he graph of f a he poi 3 whose -coordiae is 4. (b) Le k = 4, so ha f =. Deermie wheher f has a relaive miimum, a relaive maimum, or 4 eiher a =. Jusify your aswer. (c) Fid he value of k for which f has a criical poi a = 5. (d) Le k = 6, so ha f =. Fid he parial fracio decomposiio for he fucio f. 6 Fid f d (a) f( 4) = = f (4) = = A equaio for he lie age o he graph of f a he poi 5 whose -coordiae is 4 is y = ( 4 ) (b) f = f = = 4 4 f chages sig from posiive o egaive a =. Therefore, f has a relaive maimum a =. : slope : : age lie equaio : cosiders f : : aswer wih jusificaio k ( 5) (c) f ( 5) = = k = ( 5) k ( 5) (d) A B A( 6) B 6 = ( 6) = + 6 = + = = A ( 6) A = 6 = 6 = B ( 6) B = = + ( 6) f d = + d 6 6 = l + l 6 + C = l + C : aswer 5 The College Board. : parial fracio decomposiio 4: : geeral aiderivaive

7 The Maclauri series for a fucio f is give by coverges o AP CALCULUS BC 5 SCORING GUIDELINES Quesio 6 ( ) ( 3) = ad = f for < R, where R is he radius of covergece of he Maclauri series. (a) Use he raio es o fid R. (b) Wrie he firs four ozero erms of he Maclauri series for f, he derivaive of f. Epress f as a raioal fucio for < R. (c) Wrie he firs four ozero erms of he Maclauri series for. he hird-degree Taylor polyomial for g = e f abou =. e Use he Maclauri series for e o wrie (a) Le a be he h erm of he Maclauri series. + a+ ( 3) 3 = = a + ( 3) + 3 lim = < < 3 : ses up raio 3 : : compues limi of raio : deermies radius of covergece The radius of covergece is R =. 3 (b) The firs four ozero erms of he Maclauri series for f are f = = ( 3) + 3 : firs four ozero erms 3: : raioal fucio (c) The firs four ozero erms of he Maclauri series for ! 3! The produc of he Maclauri series for series for f is ( 3 ) ! 3! + 3 = + + e are e ad he Maclauri The hird-degree Taylor polyomial for g = e f 3 abou = is T3 = +. 3: : firs four ozero erms of he Maclauri series for : Taylor polyomial e 5 The College Board.