AP 微积分 AB 1999 真题与答案

AP Calculus AB 1999 Scoring Guidelines These materials were produced by Educational Testing Service (ETS), which develops and administers the examinations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle. The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in 1900, the association is composed of more than 3,900 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. 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This permission does not apply to any third-party copyrights contained herein. AB? ?Aparticlemovesalongthey?axiswithvelocitygivenbyv?t?tsin?t ? ?fort? ?a?Inwhichdirection?upordown?istheparticlemovingattimet?Why? ?b?Findtheaccelerationoftheparticleattimet?Isthevelocityoftheparticleincreasingat t?Whyorwhynot? ?c?Giventhaty?t?isthepositionoftheparticleattimetandthaty?ndy? ?d?Findthetotaldistancetraveledbytheparticlefromt?tot? ?a?v?sin? ? ? Up?becausev? ?answerandreason ?b?a?t?v ? ?t?sint ? ?t ? cost ? a?v ? ?or? No?visdecreasingat?becausev ? ? ? ? ?a? ?conclusionandreason ?c?y?t? Z v?t?dt ? Z tsint ? dt? cost ? ? ?C y? ? ? ?C?C? ? ? y?t? ? ? cost ? ? ? ? y? ? ? cos? ? ? ?or? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?y?t? Z v?t?dt ?y?t? ? ? cost ? ?C ?y? ?d?distance? Z ? ? jv?t?jdt? or v?t?tsint ? ? t?ort? p ? y?y? p ?y?or? ?y? p ?y?y? p ?y? ?or? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?limitsof?and?onanintegralof v?t?orjv?t?j or usesy?andy?tocomputedistance ?handleschangeofdirectionatstudent?s turningpoint ?answer ?ifincorrectturningpoint AB?BC? ?Theshadedregion?R?isboundedbythegraphofy?x ? andtheline y?asshowninthe?gureabove? ?a?FindtheareaofR? ?b?FindthevolumeofthesolidgeneratedbyrevolvingRaboutthe x?axis? ?c?Thereexistsanumberk?k?suchthatwhenRisrevolvedabout theliney?k?theresultingsolidhasthesamevolumeasthesolidin part?b?Write?butdonotsolve?anequationinvolvinganintegral expressionthatcanbeusedto?ndthevalueofk? O x y y 4 y x2 ?a?Area? Z ? ? ?x ? ?dx ? Z ? ? ?x ? ?dx ? ? ?x? x ? ? ? ? ? ? ? ? ?or? ? ? ?integral ?answer ?b?Volume? Z ? ? ? ? ? ?x ? ? ? ? dx ? Z ? ? ?x ? ?dx ? ? ?x? x ? ? ? ? ? ? ? ? ?or? ? ? ? ? ? ? ? ? ?limitsandconstant ?integrand ?answer ?c? Z ? ? ? ?k?x ? ? ? ?k? ? ? dx? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?limitsandconstant ?integrand ?eacherror ?equation AB?BC? ?Therateatwhichwater?owsoutofapipe?ingallonsperhour?is givenbyadi?erentiablefunctionRoftimet?Thetableabove showstherateasmeasuredevery?hoursfora?hourperiod? ?a?UseamidpointRiemannsumwith?subdivisionsofequal lengthtoapproximate Z ? ? R?t?dt?Usingcorrectunits?explain themeaningofyouranswerintermsofwater?ow? ?b?Istheresometimet?t?suchthatR ? ?t?Justify youranswer? ?c?Therateofwater?owR?t?canbeapproximatedby Q?t? ? ? ? ?t?t ? ? ?UseQ?t?toapproximatethe averagerateofwater?owduringthe?hourtimeperiod? Indicateunitsofmeasure? tR?t? ?hours?gallonsperhour? ? ? ? ? ? ? ? ? ? ?a? Z ? ? R?t?dt?R?R?R?R? ? ?gallons Thisisanapproximationtothetotal?owin gallonsofwaterfromthepipeinthe?hour period? ? ? ? ? ? ? ? ? ?R?R?R?R? ?answer ?explanation ?b?Yes? SinceR?R?theMeanValue Theoremguaranteesthatthereisat?t? suchthatR ? ?t? ? ? ?answer ?MVTorequivalent ?c?Averagerateof?ow ?averagevalueofQ?t? ? ? ? Z ? ? ? ? ?t?t ? ?dt ?gal?hror?gal?hr ? ? ? ? ? ? ? ? ?limitsandaveragevalueconstant ?Q?t?asintegrand ?answer ?units?Gallonsinpart?a?andgallons?hrin part?c?orequivalent? ?units AB? ?Supposethatthefunctionfhasacontinuoussecondderivativeforallx?andthatf?f ? ? andf ? ?Letgbeafunctionwhosederivativeisgivenbyg ? ?x?e ?x ?f?x?f ? ?x?forallx? ?a?Writeanequationofthelinetangenttothegraphoffatthepointwherex? ?b?Istheresu?cientinformationtodeterminewhetherornotthegraphoffhasapointofin?ection whenx?Explainyouranswer? ?c?Giventhatg?writeanequationofthelinetangenttothegraphofgatthepointwhere x? ?d?Showthatg ? ?x?e ?x ?f?x?f ? ?x?f ? ?x?Doesghavealocalmaximumatx?Justify youranswer? ?a?Slopeatx?isf ? ? Atx?y? y?x? ?equation ?b?No?Whetherf ? ?x?changessignatx?is unknown?Theonlygivenvalueoff ? ?x?is f ? ? ? ? ?answer ?explanation ?c?g ? ?x?e ?x ?f?x?f ? ?x? g ? ?e ? ?f?f ? ? ? y?x? y? ? ? ?g ? ? ?equation ?d?g ? ?x?e ?x ?f?x?f ? ?x? g ? ?x?e ?x ?f?x?f ? ?x? ?e ?x ?f ? ?x?f ? ?x? ?e ?x ?f?x?f ? ?x?f ? ?x? g ? ?e ? ? Sinceg ? ?andg ? ?gdoeshavea localmaximumatx? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?verifyderivative ?productorchainruleerror ?algebraerrors ?g ? ?andg ? ? ?answerandreasoning AB?BC? ?Thegraphofthefunctionf?consistingofthreelinesegments?is givenabove?Letg?x? Z x ? f?t?dt? ?a?Computeg?andg? ?b?Findtheinstantaneousrateofchangeofg?withrespecttox?at x? ?c?Findtheabsoluteminimumvalueofgontheclosedinterval ?Justifyyouranswer? ?d?Thesecondderivativeofgisnotde?nedatx?andx? Howmanyofthesevaluesarex?coordinatesofpointsof in?ectionofthegraphofg?Justifyyouranswer? O (1, 4) (2, 1) (4, 1) 112234 1 1 2 2 3 4 ?a?g? Z ? ? f?t?dt? ? ? ? ? ? ? ? ? ? ? ? g? Z ? ? f?t?dt? ? ? ? ? ? ?g? ?g? ?b?g ? ?f?answer ?c?gisincreasingon?anddecreasingon? Therefore?ghasabsoluteminimumatan endpointof? Sinceg?andg? ? ? ? theabsoluteminimumvalueis? ? ? ? ? ? ? ? ? ?interioranalysis ?endpointanalysis ?answer ?d?One?x? On?g ? ?x?f ? ?x? On?g ? ?x?f ? ?x? On?g ? ?x?f ? ?x? Therefore?g?isapointofin?ectionand ?g?isnot? ? ? ? ? ? ? ? ? ?choiceofx?only ?show?g?isapointofin?ection ?show?g?isnotapointofin?ection AB? ?Inthe?gureabove?line?istangenttothegraphofy? ? x ? atpointP?withcoordinates ? w? ? w ? ? ?wherew?Point Qhascoordinates?w?Line?crossesthex?axisatthe pointR?withcoordinates?k? ?a?Findthevalueofkwhenw? ?b?Forallw?ndkintermsofw? x y RQ 1 x2 y = P O ?c?Supposethatwisincreasingattheconstantrateof?unitspersecond?Whenw?whatisthe rateofchangeofkwithrespecttotime? ?d?Supposethatwisincreasingattheconstantrateof?unitspersecond?Whenw?whatisthe rateofchangeoftheareaof?PQRwithrespecttotime?Determinewhethertheareaisincreasing ordecreasingatthisinstant? ?a? dy dx ? ? x ? ? dy dx ? ? ? ? x? ? ? ? Line?through ? ? ? ? ? and?k?hasslope? ? ? ? Therefore? ? ? ? k? ? ? ? or? ? ? ? ? ? ?k? k? ? ? ? ? ? ? ? ? ? dy dx ? ? ? ? x? ?answer ?b?Line?through ? w? ? w ? ? and?k?hasslope? ? w ? ? Therefore? ? ? w ? k?w ? ? w ? or? ? w ? ? ? w ? ?k?w? k? ? ? w ? ? ? ? ? ? ?equationrelatingwandk? usingslopes ?answer ?c? dk dt ? ? ? dw dt ? ? ? ? ? ? ? dk dt ? ? ? ? w? ? ? ? ?answerusing dw dt ? ?d? ? ? ? ? ? ? ? ? ? ? ? ? ? ?