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YEAR122 Uni tSecondEdi t i onCAMBRIDGEMat hemat icsBI LL PENDERDAVI D SADLERJ ULI A SHEADEREK WARDCOLOURVERSION WITHSTUDENT CD- ROMNow i n col our wi t h anel ect r oni c ver si on of t he book on CDCAMBRIDGE UNIVERSITY PRESS Cambri dge, New York, Mel bourne, Madri d, Cape Town, Si ngapore, So Paul o, Del hi Cambri dge Uni versi ty Press 477 Wi l l i amstown Road, Port Mel bourne, VIC 3207, Austral i a www.cambri dge.edu.au Informati on on thi s ti tl e: www.cambri dge.org/ 9780521177504 Bi l l Pender, Davi d Sadl er, Jul i a Shea, Derek Ward 2009 Fi rst edi ti on 1999 Repri nted 2001, 2004 Second edi ti on 2005 Col our versi on 2009 Cover desi gn by Syl vi a Wi tte Typeset by Aptara Corp. Pri nted i n Chi na by Pri ntpl us Nat i onal Li br ar y of Aust r al i a Cat al ogui ng i n Publ i cat i on dat a Bi l l Pender Cambri dge mathemati cs 2 uni t : year 12 / Bi l l Pender [et al .] . 2nd ed. 9780521177504 (pbk.) Incl udes i ndex. For secondary school age. Mathemati cs. Mathemati cs--Probl ems, exerci ses, etc. Sadl er, Davi d. Shea, Jul i a. Ward, Derek. 510 ISBN 978-0-521-17750-4 paperback Reproducti on and Communi cati on f or educati onal purposes The Austral i an Copyr i ght Act 1968 (the Act) al l ows a maxi mum of one chapter or 10% of the pages of thi s publ i cati on, whi chever i s the greater, to be reproduced and/ or communi cated by any educati onal i nsti tuti on for i ts educati onal purposes provi ded that the educati onal i nsti tuti on (or the body that admi ni sters i t) has gi ven a remunerati on noti ce to Copyri ght Agency Li mi ted (CAL) under the Act. For detai l s of the CAL l i cence for educati onal i nsti tuti ons contact: Copyri ght Agency Li mi ted Level 15, 233 Castl ereagh Street Sydney NSW 2000 Tel ephone: (02) 9394 7600 Facsi mi l e: (02) 9394 7601 Emai l : i nfo@copyri ght.com.au Reproducti on and Communi cati on f or other purposes Except as permi tted under the Act (for exampl e a fai r deal i ng for the purposes of study, research, cri ti ci sm or revi ew) no part of thi s publ i cati on may be reproduced, stored i n a retri eval system, communi cated or transmi tted i n any form or by any means wi thout pri or wri tten permi ssi on. Al l i nqui ri es shoul d be made to the publ i sher at the address above. Cambri dge Uni versi ty Press has no responsi bi l i ty for the persi stence or accuracy of URLS for external or thi rd-party i nternet websi tes referred to i n thi s publ i cati on and does not guarantee that any content on such websi tes i s, or wi l l remai n, accurate or appropri ate. Informati on regardi ng pri ces, travel ti metabl es and other factual i nformati on gi ven i n thi s work are correct at the ti me of fi rst pri nti ng but Cambri dge Uni versi ty Press does not guarantee the accuracy of such i nformati on thereafter. Student CD-ROM l i cence Pl ease see the fi l e 'l i cence.txt' on the Student CD-ROM that i s packed wi th thi s book. ContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vHow to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAbout the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiChapter One Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11A Areas and the Denite Integral . . . . . . . . . . . . . . . . . . . . . . 11B The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . 61C The Denite Integral and its Properties . . . . . . . . . . . . . . . . . 121D The Indenite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 191E Finding Areas by Integration . . . . . . . . . . . . . . . . . . . . . . . 251F Areas of Compound Regions . . . . . . . . . . . . . . . . . . . . . . . . 331G Volumes of Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . 391H The Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 471I Simpsons Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511J Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 55Chapter Two The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . 602A Review of Exponential Functions . . . . . . . . . . . . . . . . . . . . . 602B The Exponential Function exand the Denition of e . . . . . . . . . . 652C Dierentiation of Exponential Functions . . . . . . . . . . . . . . . . . 732D Applications of Dierentiation . . . . . . . . . . . . . . . . . . . . . . . 792E Integration of Exponential Functions . . . . . . . . . . . . . . . . . . . 842F Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 902G Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 96Chapter Three The Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . 993A Review of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 993B The Logarithmic Function Base e . . . . . . . . . . . . . . . . . . . . . 1053C Dierentiation of Logarithmic Functions . . . . . . . . . . . . . . . . . 1123D Applications of Dierentiation of log x . . . . . . . . . . . . . . . . . . 1163E Integration of the Reciprocal Function . . . . . . . . . . . . . . . . . . 1213F Applications of Integration of 1/x . . . . . . . . . . . . . . . . . . . . . 1283G Calculus with Other Bases . . . . . . . . . . . . . . . . . . . . . . . . . 1333H Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 139r riv ContentsChapter Four The Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 1414A Radian Measure of Angle Size . . . . . . . . . . . . . . . . . . . . . . . 1414B Mensuration of Arcs, Sectors and Segments . . . . . . . . . . . . . . . 1474C Graphs of the Trigonometric Functions in Radians . . . . . . . . . . . 1534D The Behaviour of sinx Near the Origin . . . . . . . . . . . . . . . . . . 1594E The Derivatives of the Trigonometric Functions . . . . . . . . . . . . . 1644F Applications of Dierentiation . . . . . . . . . . . . . . . . . . . . . . . 1724G Integration of the Trigonometric Functions . . . . . . . . . . . . . . . . 1784H Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 1864I Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 192Chapter Five Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965A Average Velocity and Speed . . . . . . . . . . . . . . . . . . . . . . . . 1965B Velocity as a Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 2035C Integrating with Respect to Time . . . . . . . . . . . . . . . . . . . . . 2135D Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 220Chapter Six Rates and Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . 2236A Applications of APs and GPs . . . . . . . . . . . . . . . . . . . . . . . 2236B The Use of Logarithms with GPs . . . . . . . . . . . . . . . . . . . . . 2326C Simple and Compound Interest . . . . . . . . . . . . . . . . . . . . . . 2386D Investing Money by Regular Instalments . . . . . . . . . . . . . . . . . 2446E Paying O a Loan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2526F Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2606G Natural Growth and Decay . . . . .