SOLUTIONS AP Calculus Problems Part Il AB and BC 1986 — 2000 Judith Broadwin George Lenchner Martin Rudolph Mathematical Ofympiads jor Elementary aa Middle Schoois An important notice from Nicholas J. Restivo, Executive Director of Mathematical Olympiads for Elementary & Middle Schools [MOEMS] All AP” Exam questions are the copyrighted intellectual property of the College Board, which has granted permission for us to share them. SOLUTIONS AP Calculus Problems Part fl AB and BC 1986 - 2000 Judith Broadwin George Lenchner Martin Rudolph Copyright © 2000 by Mathematica! Wsmipicis for Elementary and Middls Sehaote, all righ revered ISBN 4.082144.08-2 FOREWORD . page USING THIS ROOK. page 3 UPDATING YOUK BOOK... page 4 INDRX BY TOPICS. pages 5-6 Soluton pages are identced by sea, type of eaara (AB or BC), and prablom numer, Tdemvifcahon of sach page appeurs bath al Cie Inp und hotiare of the page. The following conventions ae observe + Years ve ssted is onde Com £986 ‘brough Zo 2000, + AB sxames ane listed belure BU exems, Sram prablers | + Problems ari sofutioas for euk sham are Tinie in exe » Problems and sulutioes commun lath ihe AB sl BC examinations ars Vista under the AB dexignation Foreworp Petiodiee ly, tre CoMege Bua’ Advanced Hlncoraent ogra provdien arproved, commplers<ohutions for te Part ILproblems om AP I"yaminalions in Caleutus, Colkego Board Online aorg the lates! AP Progrart infoanation to teachers on che weh ‘Ute ailrasti: ewww cullegebnon’ orp’ap aml nd400! bal, feat, ‘over 14 publications ahant Caloulus are offeaod by sho program. Fur wr AP Publications Quder Fona ite lu: Advenced Placousent Program, ?.O. Bax 4570, Princeton, NF-UE342-66i0, To luelpseackets of Advanced Plasemen: Caloulus an choir studerts, sho Mfisheratica! Olympiads Sor Klementary aud Middle Sehacla IMOBMS), xnol-for-profit public foucdstion, spensnes the proluctive ul this Solutions Book, Tho soaks Tee each of ths thurssen years from 1982 Tseng 1995 were whom ay the le Tudilh Proudwia (fetiebo Bigh Schoo’, NY) and Ds. George Lenchner (Bx xutive Dinectot, MOEMS). Upon I, -encimersredreanest in 1996, Wianin Rudalpts (Oceanside High Schea:, NY} was added te the zen ‘The following peopte seved as covsultanis Mas. Elizabesh Aylmer, Velley Steam Ceaual High Schaal, NY; Mr, Tawi Bemaeor. Westoury Eig Schoel, NY; Ms. 2va Demyen, Valley Sieear: High Soho! Diexiet, NYS Mr. Petes Hollensicin, ‘Malverne ifigh School, NY (now retired) Mins, Sandra Leumann, Valley Streara Soul High School, WY inow retired), Dy, Musine Litshizz, Locust Valley Triends Azaderay. 61, Mr, losspa MeConnack, Whearley School, NY (apw retired); Ms. Nauey Paugh, Woodbridge Township Seana! Dintric:, NT; Mt. Marlin Radalob, Oceanside Frigh School, NY sd Mis, Shicls Sirauss, Vuater Calisge igh Sehnsl, SY ‘The manuscript was prepare Zor Rancona is respousible Zor the frst set computer salulivns ‘The book is Uisteibuted by MOBMS and MOEDIS Ati, Rican! Kalman, Bxecutive Diresloc 215¢ Bel'more Avenue Rellmore, NY 1710564 PLONE: (546)781-2400 MAI: mown 2000 com FAX: (S16)785-66900 WED SITE: wow moomsore oe Catenin problems ace from toe 1986-2000 AU & BC Patt LL AP Calcuias Foam problems tave buen reprinted oy peaulssion of Kucaional Testing Service and the College Patrance Examination Board, the copyright owners, For fimile! use by Uwe Matheuratical Olyampisds for Tlememary ard Middle Schcols, Hermniion ta senrint the piublars Joos aut eunstitace Feview or eudorsemenl by Fducational Testing Scrvice or fy tke Callege 3oard of thes publicacion a ¢ whole fr af the coiteions it comtsins b, Reirions Hinepier, ang Mestin Raph ray anil Must fo the cemalesas, jlicarion sy Nancy Pau F compet sol sequests for informetion shoul ae sent w: ins, Thsse Usinc Tris Book Some teachers ask all students to panease copy of tho Solations Hook for AP Calculus, wale others proferso have a copy just sor hemee'ves, we asked an oubsancling teacher oF AP Caicubus ty tol ve how sh uses the Sofutions Hoc, Merrespense As a iach, Tuse the Sousions Book (which stalenss purchase) as another text, The AP rablems sare so ric and varied Sat Frofer to cham as slustrarve problems all year. If students use ths hook corracsiy fxd mes! do, shey try the problem chemsehves and then check tha’ saloon wlth the book Sometimes the sient s apiruch aay be different jean that of the Boak. in cuch cases, we iisess the mers of wack This resulss im ahinking of Uifferend ways of sofeing problenes und the relative efficiency of each According 50 forowr studenc, the Solutions Hook an the notes they write én set copy serve sa vahwible nefonence when thas po an to college, (Ou the other hand, a teacher may fee! thet vhor some students are assigued AP protien, they wi copy from the Sohuions Rook, Another teachcr may argue Cal Soluions Baas are nce available, those smdens swauld cay from ota srndents ‘We suggest hat teachery explain lo students the procecaren ané aitivade that Lead to intelotual growth end development when sing a reaounce like the Solutions Rook, Ieisimupostane for students 10 heas this ‘Hrongitout thei achoolcetoet. Wena sludext weempis nsirabiem without help, the hosk"s latin shold ‘ne hidden, Upon complete. dhe slusient shoa'd compare kis or ler solution wilh real ofthe hook. ee stucens soltion is diferent from the‘sook's or has special fextures, the swacher should discuse with che fa stadsnt carmot do 2 probfom, the sudens shocld lool at Une eswer or loa atthe diagram for elaos Auuther stray is i revcal one fiuc of the buok's solution ata sme watil the smdent is able to eontiae ‘without haf. rts sc, the student should be avid to retry the problem ae ar two weeks ater ‘ako samnegy as boon resaine Used cbis way. the Solutjons Book will become an ipo resuree for tac stents geoveth ans ‘uthemedieal develop Inpex By Topics Some topics which zppear belovs meay uo Tonget be rognisod, For changes in the AP Calculus causes, ‘orale atest slianced! Placessent Course Descripron. Mathematics: Cafes 4B, Caleulys BC. Each ofthe topics liste below yives she AP Caleuins examinations ofthe most recent fifcen-sear period ‘which are velaed cathe wpie Codinge (8) 8THCIab denotes the 1987 BC exe, problem |, perisa Beh: C)IRABARCD exons that reales 4 ef the 1993 AB oxam and probes 3 of the BC exam are idemcal Anuistfferen igsion (initial conditions): SYABG SORCI2 91 Ata GOARLe Are Length, Surface Area: O1AC&t 9TBCIC 9TBCRD Area: MOABSa S7BCIu{In SRARSu EEBC2n SARC) WUABIa(e) YIABJa 9IECIA SEABSBCZ(C) BIABNBCHk S4AB2RCI ale?) 9SAHSBCI NGABZab ITABIabe ITABIBCS YTBCis VIBC4e SABIe) 98DCIy IARI} GOABRC) 4 Average Rate of Change: Y44 00 92AB30 Average Values 2GABSe S8AT5s SERChHab BIRCH BCUb MABLe IZAH2e S6ABLEC)3b Y7ABIC Cuneuvily; PeintsufTafiection: 86ABie S7ABéh E8ABMc BEBCTh SIABSh 92ABIh BCS 93ARIA SBA SCIbitn a) SAAR Ie ORCS SAR(BCMd STAB(SCISL 994BMd 99ABKECIS Continaity {Ditterentiab@ity: 8644 D2BC4 948B3e Curve Sketebing: SCAB? S8AB4ei—*) YOAD6 9OBC4a(pulse) 91 ARS STBCKe') OSABI 98AR(BCISH Deflaice Inceyral: SUC 974.8(%4:)5a Devivalive, Deliniion: SOBC6ad BORLA 99RC6ah Derivative. Graph of: S9AR5 O2ABS D4ABG YSHCG YOAB] YYABLEC)S GLABI Hifferential Bquations: S6BC4 8SRCK: 8IACS O1BCSh HDARK GRABS YIBCGC ECS STABLECIS UNAWed SERUZ UWABG UOBCGod Tatierentiotien, Implicit: SBC2a SZABABUT] S4ABIa OBARGL MATHTES Ditiereut sft,“ AaGeih: SABI ATABGbe StAIc BEBCla BVABEA SOARIHON 4) S1A8dbe SAB) # Dtauee Travulod: SYABLG DUABIC YIBCId 92AD2h QICHa(e TSARIOe" ORCI IPARLE SPABIE UCABECI2s Krror: PR CSba MECS: WECED OUACIE iat Growth: RRC SFARG SHABIEC HA BIBCE BRACK DIRCAd: MABSD O7AB3d SYAB(HCIS S9BCie HABA Improper Eniegeoks 948CIa Indegeation Methods: 875Cabtvarts) 90BCH pats} Enverval/ Convergence (see Series): STBCHh ASRC4 IIBCSe S4BCSh ORCA OMRCI YBCEY Maximum, Minimum SGA8:¢ $6ABS> STAB{a S7BC2o SABHCNCY REC:¢ SOBCA SOAK: Logeauge Error Bevud: YOECIh 9TARSa IACIn D2ARYed 92ABIc GIARABCI (Ia 7) S4AHID DMABAR(h, xy 94BCE DSAHOd ORRCTh IGARIACMe 97STda STARAC)S IRAR(AC}2 IOABAA 9PAREC)Se RC!Y D0BCaed Mean Value nud Ralles' Tueorems: S¥ABle Y3ABNBCLia STAB2e SVABIBCISD ‘Motion in a Plane (soe Vectors) Motion on a Line: SoAB3(n 2) S7ABi 8SAB2 QYABS SDAMCHC}L JIBC] SARZa VIABLE) 94AB4(In.s) 95AB2 97AB1 S8ABIac O9ABL QOAB(BCY? Parametric Eiustioan: SibCSivclocisys EBC4 DBC) OSBCL OBO FTBCL IBC 9BBCL Polar Conedinules: 901 S(areay 93002 Frecalenlus: S6ABLa(zer0s} S6AB2aLK: ty) S9AB1a_ S9ABAabc(Di}, asymptotes) 9UAH2: Rectangle Approximation Mashed (RAM): OAR. ATHNC Related Rates: #7ABS SSHC? QUAH4 S1ABG DUABG 92KCSab MABMBCZ) ISABSIBCI) SGABSe(BCS8) SPABKcd Niemann Sums: 98AB3 D9ABEOSA alive Tess 99ATAAG 99DCHE (GBC5( Tayler) $7C4z6( Tavlos} 88BC 9NRCSiTaylor] S1BCSc Taylor) OBC 93RCSTaytne) Tayler} SEBCACTayion) GORCL STECATayles} ORKCICLavlar) IRCA Taylor Polywomial) (Eaylor Polynarniai) Slope Fieldx und fuler’s Method: 9880435 99BCG5 ‘Tengemts, Norusals: S6ABID @7AB2d Q7EC2 S8ABid SIABIb soBCAd(parsmecric squstiors) SOBCSa ILABIdb 9ZABLe SIARKBCII VABSBCIn(e) OBC} VARNACTn OARIa S5ABSbe 9SAB(BC hc SABE STABZH 97ARSc Y7BCth SEAR YSABGGC YPAKAIe YU AEah s98Cba Total Change Given 2 Rate of Chonge: 9RARIBC)ML IRARLBOVSIL ‘Teapezolda! approximation: MABSa 96AS(BC}3e Yeeturs: S7BCS 92BC3e") SBC 9SBCI SEBCHE ITBCL 9BC6 MBC] DUBCH Volume of Sofid of Revolution: S6ADEA KTAU3ab STOCIbelIn si SKATES ARBOD $9A071 S9BCZ IWABIbe DONCIab SLAB. HRCI FABSLBOZiciet) BABKBCHE S4AGABCIjBele SSAB4ECDo YOABSa SOBC%e Y7AB2d SYABlod YEECIbe SYAR(BC)2be HUABCLC)T9 Voiwrus by Sting: S8BCS 9IEC3¢ S6AB2 YIRCSe WAB(RCIIS Ware. BCS I6BCSb ptotes) ATAR2ACKEY YABLALDID, same AB3e S1ARQ¢absolute valae) S2AB3Ab