www.ebookslides.com GLOBAL EDITION Precalculus Concepts Through Functions A Unit Circle Approach to Trigonometry THIRD EDITION JDIBFM4VMMJWBOr.JDIBFM4VMMJWBO*** www.ebookslides.com Precalculus Concepts Through Functions A Unit Circle Approach To Trigonometry Third Edition Global Edition Michael Sullivan Chicago State University Michael Sullivan, lll Joliet Junior College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.ebookslides.com Editor in Chief: Anne Kelly Acquisitions Editor: Dawn Murrin Assistant Editor: Joseph Colella Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Peggy McMahon Digital Assets Manager: Marianne Groth Associate Media Producer: Marielle Guiney Head, Learning Asset Acquisition, Global Edition: Laura Dent Assistant Acquisitions Editor, Global Edition: Aditee Agarwal Asistant Project Editor, Global Edition: Mrithyunjayan Nilayamgode Associate Print & Media Editor, Global Edition: Anuprova Dey Chowdhuri** Senior Manufacturing Controller, Production, Global Edition: Trudy Kimber QA Manager, Assessment Content: Marty Wright Senior Marketing Manager: Michelle Cook Marketing Manager: Peggy Sue Lucas Marketing Assistant: Justine Goulart Senior Author Support/Technology Specialist: Joe Vetere Procurement Manager: Vincent Scelta Procurement Specialist: Debbie Rossi Text Design: Tamara Newnam Production Coordination, Associate Director of Design, USHE EMSS/HSC/EDU: Andrea Nix Image Manager: Rachel Youdelman Photo Research: Integra, Inc Text Permissions Liaison Manager: Joseph Croscup Art Director: Heather Scott Cover Art: © Laborant/Shutterstock Cover Design: Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of Michael Sullivan and Michael Sullivan, III to be identified as the authors of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Precalculus: Concepts through Functions, A Unit Circle Approach to Trigonometry, 3rd edition, ISBN 978-0-321-93104-7, by Michael Sullivan and Michael Sullivan, III, published by Pearson Education © 2015 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs ISBN 10: 1-292-05874-9 ISBN 13: 978-1-292-05874-0 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 14 13 12 11 10 Typeset in Times Ten by Cenveo® Publisher Services Printed and bound by Courier Kendallville in The United States of America www.ebookslides.com For Michael S., Kevin, and Marissa (Sullivan) Shannon, Patrick, and Ryan (Murphy) Maeve, Sean, and Nolan (Sullivan) Kaleigh, Billy, and Timmy (O’Hara) The Next Generation www.ebookslides.com www.ebookslides.com Contents F To the Student 15 Preface to the Instructor 17 Prepare for Class ‘‘Read the Book’’ 21 Practice ‘‘Work the Problems’’ 22 Review ‘‘Study for Quizzes and Tests’’ 23 Resources for Success 24 Applications Index 25 Foundations: A Prelude to Functions F.1 The Distance and Midpoint Formulas 33 34 6TFUIF%JTUBODF'PSNVMBr6TFUIF.JEQPJOU'PSNVMB F Graphs of Equations in Two Variables; Intercepts; Symmetry 41 (SBQI&RVBUJPOTCZ1MPUUJOH1PJOUTr'JOE*OUFSDFQUTGSPNB(SBQIr'JOE *OUFSDFQUTGSPNBO&RVBUJPOr5FTUBO&RVBUJPOGPS4ZNNFUSZr,OPX)PX to Graph Key Equations F Lines 51 $BMDVMBUFBOE*OUFSQSFUUIF4MPQFPGB-JOFr(SBQI-JOFT(JWFOB1PJOU BOEUIF4MPQFr'JOEUIF&RVBUJPOPGB7FSUJDBM-JOFr6TFUIF1PJOU4MPQF 'PSNPGB-JOF *EFOUJGZ)PSJ[POUBM-JOFTr'JOEUIF&RVBUJPOPGB-JOF (JWFO5XP1PJOUTr8SJUFUIF&RVBUJPOPGB-JOFJO4MPQF*OUFSDFQU'PSN r*EFOUJGZUIF4MPQFBOEy*OUFSDFQUPGB-JOFGSPN*UT&RVBUJPOr(SBQI -JOFT8SJUUFOJO(FOFSBM'PSN6TJOH*OUFSDFQUTr'JOE&RVBUJPOTPG 1BSBMMFM-JOFTr'JOE&RVBUJPOTPG1FSQFOEJDVMBS-JOFT F Circles 66 8SJUFUIF4UBOEBSE'PSNPGUIF&RVBUJPOPGB$JSDMFr(SBQIB$JSDMF r8PSLXJUIUIF(FOFSBM'PSNPGUIF&RVBUJPOPGB$JSDMF Chapter Project Functions and Their Graphs 1.1 Functions 73 74 75 %FUFSNJOF8IFUIFSB3FMBUJPO3FQSFTFOUTB'VODUJPOr'JOEUIF7BMVFPGB 'VODUJPOr'JOEUIF%PNBJOPGB'VODUJPO%FGJOFECZBO&RVBUJPOr'PSN the Sum, Difference, Product, and Quotient of Two Functions 1.2 The Graph of a Function 88 *EFOUJGZUIF(SBQIPGB'VODUJPOr0CUBJO*OGPSNBUJPOGSPNPSBCPVUUIF Graph of a Function 1.3 Properties of Functions 98 %FUFSNJOF&WFOBOE0EE'VODUJPOTGSPNB(SBQIr*EFOUJGZ&WFOBOE 0EE'VODUJPOTGSPNUIF&RVBUJPOr6TFB(SBQIUP%FUFSNJOF8IFSFB 'VODUJPOJT*ODSFBTJOH %FDSFBTJOH PS$POTUBOUr6TFB(SBQIUP-PDBUF -PDBM.BYJNBBOE-PDBM.JOJNBr6TFB(SBQIUP-PDBUFUIF"CTPMVUF BYJNVNBOEUIF"CTPMVUF.JOJNVNr6TFB(SBQIJOH6UJMJUZUP"QQSPYJNBUF Local Maxima and Local Minima and to Determine Where a Function is *ODSFBTJOHPS%FDSFBTJOHr'JOEUIF"WFSBHF3BUFPG$IBOHFPGB'VODUJPO www.ebookslides.com CONTENTS 1.4 Library of Functions; Piecewise-defined Functions 110 (SBQIUIF'VODUJPOT-JTUFEJOUIF-JCSBSZPG'VODUJPOTr(SBQI Piecewise-defined Functions 1.5 Graphing Techniques: Transformations 121 (SBQI'VODUJPOT6TJOH7FSUJDBMBOE)PSJ[POUBM4IJGUTr(SBQI'VODUJPOT 6TJOH$PNQSFTTJPOTBOE4USFUDIFTr(SBQI'VODUJPOT6TJOH3FGMFDUJPOT about the x-Axis and the y-Axis 1.6 Mathematical Models: Building Functions 133 Build and Analyze Functions 1.7 Building Mathematical Models Using Variation 138 $POTUSVDUB.PEFM6TJOH%JSFDU7BSJBUJPOr$POTUSVDUB.PEFM6TJOH *OWFSTF7BSJBUJPOr$POTUSVDUB.PEFM6TJOH+PJOUPS$PNCJOFE7BSJBUJPO Chapter Review 143 Chapter Test 147 Chapter Projects 148 Linear and Quadratic Functions 2.1 Properties of Linear Functions and Linear Models 150 151 (SBQI-JOFBS'VODUJPOTr6TF"WFSBHF3BUFPG$IBOHFUP*EFOUJGZ-JOFBS 'VODUJPOTr%FUFSNJOF8IFUIFSB-JOFBS'VODUJPO*T*ODSFBTJOH %FDSFBTJOHPS$POTUBOUr'JOEUIF;FSPPGB-JOFBS'VODUJPOr#VJME-JOFBS Models from Verbal Descriptions 2.2 Building Linear Models from Data 162 %SBXBOE*OUFSQSFU4DBUUFS%JBHSBNTr%JTUJOHVJTICFUXFFO-JOFBSBOE /POMJOFBS3FMBUJPOTr6TFB(SBQIJOH6UJMJUZUP'JOEUIF-JOFPG#FTU'JU 2.3 Quadratic Functions and Their Zeros 169 'JOEUIF;FSPTPGB2VBESBUJD'VODUJPOCZ'BDUPSJOHr'JOEUIF;FSPTPGB 2VBESBUJD'VODUJPO6TJOHUIF4RVBSF3PPU.FUIPEr'JOEUIF;FSPTPGB 2VBESBUJD'VODUJPOCZ$PNQMFUJOHUIF4RVBSFr'JOEUIF;FSPTPGB2VBESBUJD 'VODUJPO6TJOHUIF2VBESBUJD'PSNVMBr'JOEUIF1PJOUPG*OUFSTFDUJPOPG 5XP'VODUJPOTr4PMWF&RVBUJPOT5IBU"SF2VBESBUJDJO'PSN 2.4 Properties of Quadratic Functions 180 (SBQIB2VBESBUJD'VODUJPO6TJOH5SBOTGPSNBUJPOTr*EFOUJGZUIF7FSUFY BOE"YJTPG4ZNNFUSZPGB2VBESBUJD'VODUJPOr(SBQIB2VBESBUJD 'VODUJPO6TJOH*UT7FSUFY "YJT BOE*OUFSDFQUTr'JOEB2VBESBUJD 'VODUJPO(JWFO*UT7FSUFYBOE0OF0UIFS1PJOUr'JOEUIF.BYJNVNPS Minimum Value of a Quadratic Function 2.5 Inequalities Involving Quadratic Functions 192 Solve Inequalities Involving a Quadratic Function 2.6 Building Quadratic Models from Verbal Descriptions and from Data 196 #VJME2VBESBUJD.PEFMTGSPN7FSCBM%FTDSJQUJPOTr#VJME2VBESBUJD Models from Data 2.7 Complex Zeros of a Quadratic Function 207 'JOEUIF$PNQMFY;FSPTPGB2VBESBUJD'VODUJPO 2.8 Equations and Inequalities Involving the Absolute Value Function 210 4PMWF"CTPMVUF7BMVF&RVBUJPOTr4PMWF"CTPMVUF7BMVF*OFRVBMJUJFT Chapter Review 216 Chapter Test 219 www.ebookslides.com CONTENTS Cumulative Review 220 Chapter Projects 221 Polynomial and Rational Functions 223 3.1 Polynomial Functions and Models 224 *EFOUJGZ1PMZOPNJBM'VODUJPOTBOE5IFJS%FHSFFr(SBQI1PMZOPNJBM 'VODUJPOT6TJOH5SBOTGPSNBUJPOTr*EFOUJGZUIF3FBM;FSPTPGB1PMZOPNJBM 'VODUJPOBOE5IFJS.VMUJQMJDJUZr"OBMZ[FUIF(SBQIPGB1PMZOPNJBM 'VODUJPOr#VJME$VCJD.PEFMTGSPN%BUB 3.2 The Real Zeros of a Polynomial Function 244 6TFUIF3FNBJOEFSBOE'BDUPS5IFPSFNTr6TF%FTDBSUFT3VMFPG4JHOTUP %FUFSNJOFUIF/VNCFSPG1PTJUJWFBOEUIF/VNCFSPG/FHBUJWF3FBM;FSPT PGB1PMZOPNJBM'VODUJPOr6TFUIF3BUJPOBM;FSPT5IFPSFNUP-JTUUIF 1PUFOUJBM3BUJPOBM;FSPTPGB1PMZOPNJBM'VODUJPOr'JOEUIF3FBM;FSPTPG B1PMZOPNJBM'VODUJPOr4PMWF1PMZOPNJBM&RVBUJPOTr6TFUIF5IFPSFNGPS #PVOETPO;FSPTr6TFUIF*OUFSNFEJBUF7BMVF5IFPSFN 3.3 Complex Zeros; Fundamental Theorem of Algebra 258 6TFUIF$POKVHBUF1BJST5IFPSFNr'JOEB1PMZOPNJBM'VODUJPOXJUI 4QFDJGJFE;FSPTr'JOEUIF$PNQMFY;FSPTPGB1PMZOPNJBM'VODUJPO 3.4 Properties of Rational Functions 264 'JOEUIF%PNBJOPGB3BUJPOBM'VODUJPOr'JOEUIF7FSUJDBM"TZNQUPUFT PGB3BUJPOBM'VODUJPOr'JOEUIF)PSJ[POUBMPS0CMJRVF"TZNQUPUFPGB Rational Function 3.5 The Graph of a Rational Function 275 "OBMZ[FUIF(SBQIPGB3BUJPOBM'VODUJPOr4PMWF"QQMJFE1SPCMFNT Involving Rational Functions 3.6 Polynomial and Rational Inequalities 290 4PMWF1PMZOPNJBM*OFRVBMJUJFTr4PMWF3BUJPOBM*OFRVBMJUJFT Chapter Review 298 Chapter Test 302 Cumulative Review 302 Chapter Projects 303 Exponential and Logarithmic Functions 4.1 Composite Functions 305 306 'PSNB$PNQPTJUF'VODUJPOr'JOEUIF%PNBJOPGB$PNQPTJUF'VODUJPO 4.2 One-to-One Functions; Inverse Functions 314 %FUFSNJOF8IFUIFSB'VODUJPO*T0OFUP0OFr%FUFSNJOFUIF*OWFSTFPGB 'VODUJPO%FGJOFECZB.BQPSB4FUPG0SEFSFE1BJSTr0CUBJOUIF(SBQIPG UIF*OWFSTF'VODUJPOGSPNUIF(SBQIPGUIF'VODUJPOr'JOEUIF*OWFSTFPGB Function Defined by an Equation 4.3 Exponential Functions 326 &WBMVBUF&YQPOFOUJBM'VODUJPOTr(SBQI&YQPOFOUJBM'VODUJPOTr%FGJOF the Number er4PMWF&YQPOFOUJBM&RVBUJPOT 4.4 Logarithmic Functions Change Exponential Statements to Logarithmic Statements and Logarithmic 4UBUFNFOUTUP&YQPOFOUJBM4UBUFNFOUTr&WBMVBUF-PHBSJUINJD&YQSFTTJPOT r%FUFSNJOFUIF%PNBJOPGB-PHBSJUINJD'VODUJPOr(SBQI-PHBSJUINJD 'VODUJPOTr4PMWF-PHBSJUINJD&RVBUJPOT 343 www.ebookslides.com CONTENTS 4.5 Properties of Logarithms 356 8PSLXJUI1SPQFSUJFTPG-PHBSJUINTr8SJUFB-PHBSJUINJD&YQSFTTJPOBT B4VNPS%JGGFSFODFPG-PHBSJUINTr8SJUFB-PHBSJUINJD&YQSFTTJPOBTB 4JOHMF-PHBSJUINr&WBMVBUFB-PHBSJUIN8IPTF#BTF*T/FJUIFS/PSe r(SBQIB-PHBSJUINJD'VODUJPO8IPTF#BTF*T/FJUIFS/PSe 4.6 Logarithmic and Exponential Equations 365 4PMWF-PHBSJUINJD&RVBUJPOTr4PMWF&YQPOFOUJBM&RVBUJPOTr4PMWF Logarithmic and Exponential Equations Using a Graphing Utility 4.7 Financial Models 371 %FUFSNJOFUIF'VUVSF7BMVFPGB-VNQ4VNPG.POFZr$BMDVMBUF&GGFDUJWF 3BUFTPG3FUVSOr%FUFSNJOFUIF1SFTFOU7BMVFPGB-VNQ4VNPG.POFZ r%FUFSNJOFUIF3BUFPG*OUFSFTUPSUIF5JNF3FRVJSFEUP%PVCMFB-VNQ Sum of Money 4.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 381 'JOE&RVBUJPOTPG1PQVMBUJPOT5IBU0CFZUIF-BXPG6OJOIJCJUFE(SPXUI r'JOE&RVBUJPOTPG1PQVMBUJPOT5IBU0CFZUIF-BXPG%FDBZr6TF /FXUPOT-BXPG$PPMJOHr6TF-PHJTUJD.PEFMT 4.9 Building Exponential, Logarithmic, and Logistic Models from Data 391 #VJMEBO&YQPOFOUJBM.PEFMGSPN%BUBr#VJMEB-PHBSJUINJD.PEFMGSPN %BUBr#VJMEB-PHJTUJD.PEFMGSPN%BUB Chapter Review 399 Chapter Test 404 Cumulative Review 405 Chapter Projects 406 Trigonometric Functions 407 5.1 Angles and Their Measures 408 Convert between Decimals and Degrees, Minutes, Seconds Measures for "OHMFTr'JOEUIF-FOHUIJGBO"SDPGB$JSDMFr$POWFSUGSPN%FHSFFT UP3BEJBOTBOEGSPN3BEJBOTUP%FHSFFTr'JOEUIF"SFBPGB4FDUPSPGB $JSDMFr'JOEUIF-JOFBS4QFFEPGBO0CKFDU5SBWFMJOHJO$JSDVMBS.PUJPO 5.2 Trigonometric Functions: Unit Circle Approach 422 Find the Exact Values of the Trigonometric Functions Using a Point on the 6OJU$JSDMFr'JOEUIF&YBDU7BMVFTPGUIF5SJHPOPNFUSJD'VODUJPOTPG 2VBESBOUBM"OHMFTr'JOEUIF&YBDU7BMVFTPGUIF5SJHPOPNFUSJD Functions of p/4 = 45°r'JOEUIF&YBDU7BMVFTPGUIF5SJHPOPNFUSJD Functions of p/6 = 30° and p/3 = 60°r'JOEUIF&YBDU7BMVFTPGUIF Trigonometric Functions for Integer Multiples of p/6 = 30°, p/4 = 45°, and p/3 = 60°r6TFB$BMDVMBUPSUP"QQSPYJNBUFUIF7BMVFPGB5SJHPOPNFUSJD 'VODUJPOr6TFB$JSDMFPG3BEJVTr to Evaluate the Trigonometric Functions 5.3 Properties of the Trigonometric Functions 439 Determine the Domain and the Range of the Trigonometric Functions r%FUFSNJOFUIF1FSJPEPGUIF5SJHPOPNFUSJD'VODUJPOTr%FUFSNJOFUIF4JHOT PGUIF5SJHPOPNFUSJD'VODUJPOTJOB(JWFO2VBESBOUr'JOEUIF7BMVFTPG UIF5SJHPOPNFUSJD'VODUJPOT6TJOH'VOEBNFOUBM*EFOUJUJFTr'JOEUIF&YBDU 7BMVFTPGUIF5SJHPOPNFUSJD'VODUJPOTPGBO"OHMF(JWFO0OFPGUIF 'VODUJPOTBOEUIF2VBESBOUPGUIF"OHMFr6TF&WFO0EE1SPQFSUJFTUP Find the Exact Values of the Trigonometric Functions 5.4 Graphs of the Sine and Cosine Functions Graph Functions of the Form y = A sin (vx) Using Transformations r(SBQI'VODUJPOTPGUIF'PSNy = A cos (vx) Using Transformations 452 www.ebookslides.com CONTENTS r%FUFSNJOFUIF"NQMJUVEFBOE1FSJPEPG4JOVTPJEBM'VODUJPOTr(SBQI 4JOVTPJEBM'VODUJPOT6TJOH,FZ1PJOUTr'JOEBO&RVBUJPOGPSB4JOVTPJEBM Graph 5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 467 Graph Functions of the Form y = A tan(vx) + B and y = A cot (vx) + B r(SBQI'VODUJPOTPGUIF'PSNy = A csc (vx) + B and y = A sec (vx) + B 5.6 Phase Shift; Sinusoidal Curve Fitting 475 Graph Sinusoidal Functions of the Form y = A sin (vx - f) + B r#VJME4JOVTPJEBM.PEFMTGSPN%BUB Chapter Review 486 Chapter Test 492 Cumulative Review 492 Chapter Projects 493 Analytic Trigonometry 6.1 The Inverse Sine, Cosine, and Tangent Functions 495 496 'JOEUIF&YBDU7BMVFPGBO*OWFSTF4JOF'VODUJPOr'JOEBO"QQSPYJNBUF 7BMVFPGBO*OWFSTF4JOF'VODUJPOr6TF1SPQFSUJFTPG*OWFSTF'VODUJPOT UP'JOE&YBDU7BMVFTPG$FSUBJO$PNQPTJUF'VODUJPOTr'JOEUIF*OWFSTF 'VODUJPOPGB5SJHPOPNFUSJD'VODUJPOr4PMWF&RVBUJPOT*OWPMWJOH*OWFSTF Trigonometric Functions 6.2 The Inverse Trigonometric Functions (Continued) 508 Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, BOE5BOHFOU'VODUJPOTr%FGJOFUIF*OWFSTF4FDBOU $PTFDBOU BOE $PUBOHFOU'VODUJPOTr6TFB$BMDVMBUPSUP&WBMVBUFTFD -1 x, csc -1 x, and cot -1 xr8SJUFB5SJHPOPNFUSJD&YQSFTTJPOBTBO"MHFCSBJD&YQSFTTJPO 6.3 Trigonometric Equations 514 4PMWF&RVBUJPOT*OWPMWJOHB4JOHMF5SJHPOPNFUSJD'VODUJPOr4PMWF 5SJHPOPNFUSJD&RVBUJPOT6TJOHB$BMDVMBUPSr4PMWF5SJHPOPNFUSJD &RVBUJPOT2VBESBUJDJO'PSNr4PMWF5SJHPOPNFUSJD&RVBUJPOT6TJOH 'VOEBNFOUBM*EFOUJUJFTr4PMWF5SJHPOPNFUSJD&RVBUJPOT6TJOHB Graphing Utility 6.4 Trigonometric Identities 523 6TF"MHFCSBUP4JNQMJGZ5SJHPOPNFUSJD&YQSFTTJPOTr&TUBCMJTI*EFOUJUJFT 6.5 Sum and Difference Formulas 531 6TF4VNBOE%JGGFSFODF'PSNVMBTUP'JOE&YBDU7BMVFTr6TF4VNBOE %JGGFSFODF'PSNVMBTUP&TUBCMJTI*EFOUJUJFTr6TF4VNBOE%JGGFSFODF 'PSNVMBT*OWPMWJOH*OWFSTF5SJHPOPNFUSJD'VODUJPOTr4PMWF Trigonometric Equations Linear in Sine and Cosine 6.6 Double-angle and Half-angle Formulas 543 6TF%PVCMFBOHMF'PSNVMBTUP'JOE&YBDU7BMVFTr6TF%PVCMFBOHMF 'PSNVMBTUP&TUBCMJTI*EFOUJUJFTr6TF)BMGBOHMF'PSNVMBTUP'JOE&YBDU Values 6.7 Product-to-Sum and Sum-to-Product Formulas 553 &YQSFTT1SPEVDUTBT4VNTr&YQSFTT4VNTBT1SPEVDUT Chapter Review 557 Chapter Test 561 Cumulative Review 561 Chapter Projects 562 www.ebookslides.com ANSWERS Section 11.4 AN95 Historical Problems (page 875) 1 1 loaves, 10 loaves, 20 loaves, 29 loaves, 38 loaves 6 (a) person (b) 2401 kittens (c) 2800 11.3 Assess Your Understanding (page 876) a - r 11 r = ; a1 = - , a2 2 17 r = ; t = , t = 2 geometric divergent series True False True r = 3; s1 = 3, s2 = 9, s3 = 27, s4 = 81 3 1 = - , a3 = - , a4 = 13 r = 2; c1 = , c2 = , c3 = 1, c4 = 15 r = 21/3; e1 = 21/3 , e2 = 22/3 , e3 = 2, e4 = 24/3 16 27 n-1 # , t = , t4 = 19 a5 = 162; an = 21 a5 = 5; an = # ( - 1)n - 23 a5 = 0; an = 16 1 n-1 n-2 29 a9 = 31 a8 = 0.00000004 33 an = # 2n - 35 an = - # a - b 25 a5 = 22; an = ( 22)n 27 a7 = = a- b 64 3 n n-1 n-1 n n n 39 an = (15) = # 15 41 - (1 - ) 43 c - a b d 45 - 37 an = - ( - 3) 15 51 47 49 53 Converges; 55 Converges; 16 69 Arithmetic; d = 1; 1375 57 Converges; 71 Neither 59 Diverges 61 Converges; 73 Arithmetic; d = - ; - 700 79 Geometric; r = - 2; - - - 22 50 23 1/2 81 Geometric; r = ; 11 + 232 11 - 325 2 83 - 75 Neither 20 63 Diverges 65 Converges; 77 Geometric; r = 18 67 Converges; 50 2 ; 2c - a b d 3 85 $39,392.81 87 (a) 0.775 ft (b) 8th (c) 15.88 ft (d) 20 ft 89 $349,496.41 91 $96,885.98 93 $305.10 95 1.845 * 1019 97 10 99 $72.67 per share 101 December 20, 2014; $9999.92 103 Option A results in a higher salary in years ($25,250 versus $24,761); option B results in a higher 5-year total ($116,801 versus $112,742) 105 Option results in the most: $16,038,304; option results in the least: $14,700,000 107 Yes A constant sequence is both arithmetic and geometric For example, 3, 3, 3, is an arithmetic sequence with a1 = and d = and is a y2 15 x2 = 114 54 geometric sequence with a = and r = 111 2.121 112 i j 113 17 17 12 11.4 Assess Your Understanding (page 882) (I) n = 1: 2(1) = and 1(1 + 1) = (II) If + + + g + 2k = k(k + 1), then + + + g + 2k + 2(k + 1) = (2 + + + g + 2k) + 2(k + 1) = k(k + 1) + 2(k + 1) = k + 3k + = (k + 1)(k + 2) = (k + 1)[(k + 1) + 1] 1 (I) n = 1: + = and (1)(1 + 5) = (6) = 2 (II) If + + + g + (k + 2) = k(k + 5), then + + + g + (k + 2) + [(k + 1) + 2] 1 1 = [3 + + + g + (k + 2)] + (k + 3) = k(k + 5) + k + = (k + 7k + 6) = (k + 1)(k + 6) = (k + 1)[(k + 1) + 5] 2 2 1 (I) n = 1: 3(1) - = and (1)[3(1) + 1] = (4) = 2 (II) If + + + g + (3k - 1) = k(3k + 1), then + + + g + (3k - 1) + [3(k + 1) - 1] 1 = [2 + + + g + (3k - 1)] + (3k + 2) = k(3k + 1) + (3k + 2) = (3k + 7k + 4) = (k + 1)(3k + 4) 2 = (k + 1)[3(k + 1) + 1] (I) n = 1: 21 - = and 21 - = (II) If + + 22 + g + 2k - = 2k - 1, then + + 22 + g + 2k - + 2(k + 1) - = (1 + + 22 + g + 2k - 1) + 2k = 2k - + 2k = 2(2k) - = 2k + - 1 (I) n = 1: 41 - = and (41 - 1) = (3) = 3 k k-1 = (4 - 1), then + + 42 + g + 4k - + 4(k + 1) - = (1 + + 42 + g + 4k - 1) + 4k (II) If + + + g + k k 1 k = (4 - 1) + = [4 - + 3(4k)] = [4(4k) - 1] = (4k + - 1) 3 3 AN96 www.ebookslides.com ANSWERS Section 11.4 1 1 = and = 1#2 + 1 k 1 1 1 = , then # + # + # + g + + (II) If # + # + # + g + 2 3 k(k + 1) k + 1 2 3 k(k + 1) (k + 1)[(k + 1) + 1] k(k + 2) + 1 1 k 1 = c # + # + # + g + d + = + = 2 3 k(k + 1) (k + 1)(k + 2) k + (k + 1)(k + 2) (k + 1)(k + 2) (k + 1)2 k + k + k + 2k + = = = = (k + 1)(k + 2) (k + 1)(k + 2) k + (k + 1) + 11 (I) n = 1: 1# # # = 2 (II) If + + + g + k = k(k + 1)(2k + 1), then 12 + 22 + 32 + g + k + (k + 1)2 1 = (12 + 22 + 32 + g + k 2) + (k + 1)2 = k(k + 1)(2k + 1) + (k + 1)2 = (2k + 9k + 13k + 6) 6 1 = (k + 1)(k + 2)(2k + 3) = (k + 1)[(k + 1) + 1][2(k + 1) + 1] 6 13 (I) n = 1: 12 = and 1 15 (I) n = 1: - = and (1)(9 - 1) = # = 2 (II) If + + + g + (5 - k) = k(9 - k), then + + + g + (5 - k) + [5 - (k + 1)] 1 = [4 + + + g + (5 - k)] + - k = k(9 - k) + - k = (9k - k + - 2k) = ( - k + 7k + 8) 2 1 = (k + 1)(8 - k) = (k + 1)[9 - (k + 1)] 2 17 (I) n = 1: # (1 + 1) = and # # # = (II) If # + # + # + g + k(k + 1) = k(k + 1)(k + 2), then # + # + # + g + k(k + 1) + (k + 1)[(k + 1) + 1] = [1 # + # + # + g + k(k + 1)] + (k + 1)(k + 2) 1 1 = k(k + 1)(k + 2) + # 3(k + 1)(k + 2) = (k + 1)(k + 2)(k + 3) = (k + 1)[(k + 1) + 1][(k + 1) + 2] 3 3 19 (I) n = 1: 12 + = 2, which is divisible by (II) If k + k is divisible by 2, then (k + 1)2 + (k + 1) = k + 2k + + k + = (k + k) + 2k + Since k + k is divisible by 2, and 2k + is divisible by 2, then (k + 1)2 + (k + 1) is divisible by 21 (I) n = 1: 12 - + = 2, which is divisible by (II) If k - k + is divisible by 2, then (k + 1)2 - (k + 1) + = k + 2k + - k - + = (k - k + 2) + 2k Since k - k + is divisible by 2, and 2k is divisible by 2, then (k + 1)2 - (k + 1) + is divisible by 23 (I) n = 1: If x 1, then x1 = x (II) Assume, for an arbitrary natural number k, that if x then xk Multiply both sides of the inequality xk by x If x 1, then xk + x 25 (I) n = 1: a - b is a factor of a1 - b1 = a - b (II) If a - b is a factor of ak - bk, then ak + - bk + = a(ak - bk) + bk(a - b) Since a - b is a factor of ak - bk and a - b is a factor of a - b, then a - b is a factor of ak + - bk + 27 (I) n = 1: (1 + a)1 = + a Ú + # a (II) Assume that there is an integer k for which the inequality holds So (1 + a)k Ú + ka We need to show that (1 + a)k + Ú + (k + 1)a (1 + a)k + = (1 + a)k (1 + a) Ú (1 + ka)(1 + a) = + ka2 + a + ka = + (k + 1)a + ka2 Ú + (k + 1)a + g + 2k = k + k + 2, then + + + g + 2k + 2(k + 1) + + + g + 2k) + 2k + = k + k + + 2k + = k + 3k + = (k + 2k + 1) + (k + 1) + + 1)2 + (k + 1) + 2 and 12 + + = The fact is that + + + g + 2n = n2 + n, not n2 + n + (Problem 1) # (1 - 1) 31 (I) n = 1: [a + (1 - 1)d] = a and # a + d = a k(k - 1) (II) If a + (a + d) + (a + 2d) + g + [a + (k - 1)d] = ka + d , then k(k - 1) a + (a + d) + (a + 2d) + g + [a + (k - 1)d] + [a + ((k + 1) - 1)d] = ka + d + a + kd k(k - 1) + 2k (k + 1)(k) (k + 1)[(k + 1) - 1] = (k + 1)a + d = (k + 1)a + d = (k + 1)a + d 2 35 {251} 36 Left: 448.3 kg; right: 366.0 kg 33 (I) n = 3: The sum of the angles of a triangle is (3 - 2) # 180° = 180° 1 (II) Assume for some k Ú that the sum of the angles of a 37 x = , y = - 3; a , - b 2 convex polygon of k sides is (k - 2) # 180° A convex polygon k sides -3 of k + sides consists of a convex polygon of k sides plus 38 c d -7 a triangle (see the illustration) The sum of the angles is k ؉ sides 29 If + + = (2 = (k But # = (k - 2) # 180° + 180° = (k - 1) # 180° = [(k + 1) - 2] # 180° www.ebookslides.com ANSWERS Chapter Test AN97 11.5 Assess Your Understanding (page 888) Pascal triangle 1; n False Binomial Theorem 10 21 50 11 13 ≈1.8664 * 1015 15 ≈1.4834 * 1013 17 x5 + 5x4 + 10x3 + 10x2 + 5x + 19 x6 - 12x5 + 60x4 - 160x3 + 240x2 - 192x + 64 21 81x4 + 108x3 + 54x2 + 12x + 23 x10 + 5x8y2 + 10x6y4 + 10x4y6 + 5x2y8 + y10 25 x3 + 22x5/2 + 30x2 + 40 22x3/2 + 60x + 24 22x1/2 + 27 a5x5 + 5a4bx4y + 10a3b2x3y2 + 10a2b3x2y3 + 5ab4xy4 + b5y5 29 17,010 31 - 101,376 33 41,472 35 2835x3 37 314,928x7 39 495 41 3360 43 1.00501 n # (n - 1)! n n! n! n n! n! n! 45 a b = = = = n; a b = = = = n - (n - 1)! [n - (n - 1)]! (n - 1)! 1! (n - 1)! n n! (n - n)! n!0! n! n n n n n n 47 2n = (1 + 1)n = a b 1n + a b (1)n - 1(1) + g + a b 1n = a b + a b + g + a b n n 51 e f ≈ 58.827 52 (a) (b) 90° (c) Orthogonal 2n - 2n 53 x = 1, y = 3, z = - 2; (1, 3, - 2) y 54 Bounded 49 (0, 6) (4, 2) (0, 0) (5, 0) x Review Exercises (page 891) a1 = - , a2 = , a3 = - , a4 = , a5 = 16 , a = , a5 = 27 13 n a1 = 8, a2 = 19, a3 = 41, a4 = 85, a5 = 173 + 10 + 14 + 18 = 48 a - 12 k + Arithmetic; d = 1; Sn = 1n + 112 Neither k k=1 n 1 n Geometric; r = 8; Sn = 18 - 12 10 Arithmetic; d = 4; Sn = 2n 1n - 12 11 Geometric; r = ; Sn = c - a b d 12 Neither 13 3825 2 1 14 9515 15 - 1320 16 a1 - 10 b ≈ 0.49999 17 35 18 12 19 22 20 5an = 53n + 21 5an = 5n - 10 22 Converges; 2 23 Converges; 24 Diverges 25 Converges; 3#1 26 (I) n = 1: # = and (1 + 12 = 3k 1k + 12, then + + + g + 3k + 1k + 12 = 13 + + + g + 3k2 + 13k + 32 (II) If + + + g + 3k = 1k + 12 3k 3k 3k 6k = 1k + 12 + 13k + 32 = + + + = 1k + 3k + 22 = 1k + 12 1k + 22 = 1k + 12 + 2 2 2 2 c1 = 2, c2 = 1, c3 = 32 , c = 1, c5 = 25 a1 = 3, a2 = 2, a3 = 27 (I) n = 1: # 31 - = and 31 - = (II) If + + 18 + g + # 3k - = 3k - 1, then + + 18 + g + # 3k - + # 31k + 12 - = 12 + + 18 + g + # 3k - + # 3k = 3k - + # 3k = # 3k - = 3k + - 1 28 (I) n = 1: 13 # - 22 = and # # 112 - 112 - = 2 2 (II) If + + + g + 13k - 22 = k 16k - 3k - 12, then 12 + 42 + 72 + g + 13k - 22 + 3 1k + 12 - 2 1 = 312 + 42 + 72 + g + 13k - 22 + 13k + 12 = k 16k - 3k - 12 + 13k + 12 = 16k - 3k - k2 + 19k + 6k + 12 2 1 2 = 16k + 15k + 11k + 22 = 1k + 12 16k + 9k + 22 = 1k + 12 1k + 12 - 1k + 12 - 2 29 10 30 32x5 + 240x4 + 720x3 + 1080x2 + 810x + 243 31 81x4 - 432x3 + 864x2 - 768x + 256 32 144 33 6048 34 (a) bricks (b) 1100 bricks 3 135 n 35 360 tiles 36 (a) 20 a b = ft (b) 20 a b ft (c) 13 times (d) 140 ft 37 $151,873.77 16 38 $23,397.17 Chapter Test (page 892) 0, 24 , , , 10 11 13 4, 14, 44, 134, 404 Geometric; r = 4; Sn = (1 - 4n) 3 - 61 + = 36 - 10 k + a ( - 1)k a b Neither k + k=1 n Arithmetic; d = - ; Sn = (27 - n) 14 73 308 680 = 27 81 81 Arithmetic: d = - 8; Sn = n(2 - 4n) AN98 www.ebookslides.com ANSWERS Chapter Test 10 Geometric; r = 125 n ;S = c1 - a b d n 11 Neither 14 First show that the statement holds for n = a1 + 12 Converges; 1024 13 243m5 + 810m4 + 1080m3 + 720m2 + 240m + 32 b = + = The equality is true for n = 1, so Condition I holds Next, assume that a1 + 1 1 b a1 + b a1 + b g a1 + b = n + is true for some k, and determine whether the formula then holds for k + Assume that n 1 1 1 1 a1 + b a1 + b a1 + b g a1 + b = k + Now show that a1 + b a1 + b a1 + b g a1 + b a1 + b k k k + = (k + 1) + = k + Do this as follows: a1 + 1 1 1 1 1 b a1 + b a1 + b g a1 + b a1 + b = c a1 + b a1 + b a1 + b g a1 + b d a1 + b k k + 1 k k + 1 b (induction assumption) = (k + 1) # + (k + 1) # = k + + = k + = (k + 1) a1 + k + k + Condition II also holds Thus, the formula holds true for all natural numbers 15 After 10 years, the car will be worth $6103.11 16 The weightlifter will have lifted a total of 8000 pounds after sets Cumulative Review (page 892) { - 3, 3, - 3i, 3i} (a) (b) e a y B - + 13601 - + 23601 - + 13601 - + 23601 , b, a , bf 18 B 18 (c) The circle and the parabola intersect at ؊4 x a B - + 13601 - + 23601 - + 13601 - + 23601 b, a b , , 18 B 18 6x + (d) e x ` x ≠ f e ln a b f y = 5x - 10 1x + 12 + 1y - 22 = 25 (a) (b) 13 (c) 2x - y2 2x x2 -1 -1 (g) g (x) = (x - 1); all reals (h) f (x) = ; {x͉ x ≠ 3} + = (x + 1) = 4(y - 2) x - 16 (e) 7x - x - (f) {x͉ x ≠ 2} CHAPTER 12 Counting and Probability 12.1 Assess Your Understanding (page 899) subset; ⊆ finite n 1A2 + n 1B2 - n 1AxB2 True ∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c, d} 11 25 13 40 15 25 17 37 19 18 21 23 15 different arrangements 25 9000 numbers 27 175; 125 29 (a) 15 (b) 15 (c) 15 (d) 25 (e) 40 31 (a) 13.6 million (b) 78.9 million 33 480 portfolios y 37 2, 5, - 38 163.9 ft 39 - 14i + 11j + 13k 36 (x22)2 (y11)2 26 x 26 12.2 Assess Your Understanding (page 906) n! n! 30 24 11 13 1680 15 28 17 35 19 21 10,400,600 1n - r2! 1n - r2!r! 23 5abc, abd, abe, acb, acd, ace, adb, adc, ade, aeb, aec, aed, bac, bad, bae, bca, bcd, bce, bda, bdc, bde, bea, bec, bed, cab, cad, cae, cba, cbd, cbe, cda, cdb, cde, cea, ceb, ced, dab, dac, dae, dba, dbc, dbe, dca, dcb, dce, dea, deb, dec, eab, eac, ead, eba, ebc, ebd, eca, ecb, ecd, eda, edb, edc 6; 60 25 5123, 124, 132, 134, 142, 143, 213, 214, 231, 234, 241, 243, 312, 314, 321, 324, 341, 342, 412, 413, 421, 423, 431, 432 6; 24 27 5abc, abd, abe, acd, ace,ade, bcd, bce, bde, cde 6; 10 29 5123, 124, 134, 234 6; 31 16 33 35 24 37 60 39 18,278 41 35 43 1024 45 120 47 132,860 49 336 51 90,720 53 (a) 63 (b) 35 (c) 55 1.157 * 1076 57 362,880 59 660 61 15 63 (a) 125,000; 117,600 (b) A better name for a combination lock would be a permutation lock, because the order of the numbers matters 67 10 sq units permutation combination 68 (g ∘ f )(x) = 4x2 - 2x - 69 sin 75° = Historical Problem (page 916) 22 + 26 22 + 26 ; cos 15° = 22 + 13 or cos 15° = 4 70 a5 = 80 (a) 5AAAA, AAAB, AABA, AABB, ABAA, ABAB, ABBA, ABBB, BAAA, BAAB, BABA, BABB, BBAA, BBAB, BBBA, BBBB6; 11 P(A wins) = ; P(B wins) = 16 16 C14, 22 + C14, 32 + C14, 42 C14, 32 + C14, 42 + + 11 + (b) P1A wins2 = = = ; P1B wins2 = = = ; the results are the same 16 16 16 16 24 24 www.ebookslides.com ANSWERS Cumulative Review AN99 12.3 Assess Your Understanding (page 916) equally likely complement F T 0, 0.01, 0.35, Probability model Not a probability model 1 1 11 S = 5HH, HT, TH, TT 6; P1HH2 = , P1HT2 = , P1TH2 = , P1TT2 = 4 4 13 S = 5HH1, HH2, HH3, HH4, HH5, HH6, HT1, HT2, HT3, HT4, HT5, HT6, TH1, TH2, TH3, TH4, TH5, TH6, TT1, TT2, TT3, TT4, TT5, TT6 6; each outcome has the probability of 24 15 S = 5HHH, HHT, HTH, HTT, THH, THT, TTH, TTT6; each outcome has the probability of 17 S = Yellow, Red, Green, Yellow, Red, Green, Yellow, Red, Green, Yellow, Red, Green 6; each outcome has the probability 1 1 of ; thus, P12 Red2 + P14 Red2 = + = 12 12 12 19 S = Yellow Forward, Yellow Backward, Red Forward, Red Backward, Green Forward, Green Backward, Yellow Forward, Yellow Backward, Red Forward, Red Backward, Green Forward, Green Backward, Yellow Forward, Yellow Backward, Red Forward, Red Backward, Green Forward, Green Backward, Yellow Forward, Yellow Backward, Red Forward, Red Backward, Green Forward, 1 1 Green Backward ; each outcome has the probability of ; thus, P11 Red Backward2 + P11 Green Backward2 = + = 24 24 24 12 21 S = 511 Red, 11 Yellow, 11 Green, 12 Red, 12 Yellow, 12 Green, 13 Red, 13 Yellow, 13 Green, 14 Red, 14 Yellow, 14 Green, 21 Red, 21 Yellow, 21 Green, 22 Red, 22 Yellow, 22 Green, 23 Red, 23 Yellow, 23 Green, 24 Red, 24 Yellow, 24 Green, 31 Red, 31 Yellow, 31 Green, 32 Red, 32 Yellow, 32 Green, 33 Red, 33 Yellow, 33 Green, 34 Red, 34 Yellow, 34 Green, 41 Red, 41 Yellow, 41 Green, 42 Red, 42 Yellow, 42 Green, 43 Red, 43 Yellow, 43 Green, n 1E2 1 = = 44 Red, 44 Yellow, 44 Green6; each outcome has the probability of ; thus, E = 522 Red, 22 Green, 24 Red, 24 Green 6; P1E2 = 48 n 1S2 48 12 1 1 23 A, B, C, F 25 B 27 P1H2 = ; P1T2 = 29 P112 = P132 = P152 = ; P122 = P142 = P162 = 31 33 35 37 5 9 10 1 17 11 41 43 45 0.55 47 0.70 49 0.30 51 0.88 53 0.67 55 0.936 57 59 61 63 65 39 18 20 20 10 25 25 67 (a) 0.57 (b) 0.95 (c) 0.83 (d) 0.38 (e) 0.29 (f) 0.05 (g) 0.78 (h) 0.71 69 (a) (b) 71 0.167 73 0.000033069 33 33 74 2; left; 3; down 75 ( - 3, 23) 76 {22} 77 (2, - 3, - 1) Review Exercises (page 920) ∅, 5Dave 6, 5Joanne 6, 5Erica 6, 5Dave, Joanne 6, 5Dave, Erica 6, 5Joanne, Erica 6, 5Dave, Joanne, Erica 17 24 29 34 7 45 25 10 336 11 56 12 48 13 128 14 3024 15 1680 16 1820 17 1,600,000 18 216,000 19 256 (allowing numbers with initial zeros, such as 011) 20 2522520 21 (a) 381,024 (b) 1260 22 (a) 8.634628387 * 1045 (b) 0.6531 (c) 0.3469 23 (a) 0.081 (b) 0.919 24 0.13 25 0.5; 0.24 26 (a) 0.68 (b) 0.58 (c) 0.32 Chapter Test (page 921) 22 3 45 5040 151,200 462 There are 54,264 ways to choose different colors from the 21 available colors There are 840 distinct arrangements of the letters in the word REDEEMED 10 There are 56 different exacta bets for an 8-horse race 11 There are 155,480,000 possible license plates using the new format 12 (a) 0.95 (b) 0.30 13 (a) 0.25 (b) 0.55 1250 14 0.19 15 P1win on $1 play2 = ≈ 0.0000000083 16 P1exactly fours2 = ≈ 0.1608 120,526,770 7776 Cumulative Review (page 922) e 1 22 22 i, + if 3 3 y 10 (؊5, 0) (1, 0) 10 x (0, ؊5) (؊2, ؊9) x ؍؊2 b - 27 27 + i, - i, - , r 2 2 y 10 (1, 6) y؍5 x Domain: all real numbers Range: 5y͉ y Horizontal asymptote: y = 5x͉ 3.99 … x … 4.01 or 33.99, 4.01 y (؊2, ؊2) x (0, ؊2) (؊1, ؊4) 8 e f 11 y x = 2, y = - 5, z = 3 P x 10 125; 700 12 a ≈ 6.09, B ≈ 31.9°, C ≈ 108.1°; area ≈ 14.46 square units 13 ≈ 0.00000000571; 175,223,510 The new format has decreased the probability of winning the jackpot www.ebookslides.com AN100 ANSWERS Section 13.1 CHAPTER 13 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 13.1 Assess Your Understanding (page 927) lim f 1x2 xSc 23 does not exist True False 25 y 15 32 11 13 27 y 15 17 19 29 y 21 Does not exist y 1.25 31 P x y 2.5 x 2.5 x x x lim f 1x2 = 13 lim f 1x2 = - xS4 33 35 y 37 lim f 1x2 = - lim f 1x2 = xS0 41 y y 5 x x lim f 1x2 does not exist xS1 47 49 d = 25; M = (4, - 7) 45 1.6 39 x xS0 lim f 1x2 = x S p/2 y 5 x x S -1 lim f 1x2 = x S -3 y 5 x 43 0.67 lim f 1x2 = xS2 lim f 1x2 = lim f 1x2 = xS0 xS0 50 Center: (2, - 1); foci: (2, - 3), (2, 1); vertices: (2, 213 - 1), (2, - 213 - 1) 2p 52 ¢ 4, ≤ 51 $7288.48 13.2 Assess Your Understanding (page 934) c product b 33 35 37 57 39 41 43 58 g - 1(x) = y 12 3 True False f (x) x 1x 11 False 45 47 - x x - 11 - 10 51 - 49 59 60° or p red 13 80 53 15 55 17 19 - 21 23 25 - 27 32 29 31 60 x4 + 8x3 + 24x2 + 32x + 16 3x 23 212 13.3 Assess Your Understanding (page 940) one-sided lim+ f 1x2 = R xSc continuous; c 10 False 13 5x͉ - … x - or - 6 x or x … 6 11 True 12 True 15 - 8, - 5, - 17 f - 82 = 0; f - 42 = 19 q 21 23 25 Limit exists; 27 No 29 Yes 31 No 33 35 37 39 41 3 43 45 Continuous 47 Continuous 49 Not continuous 51 Not continuous 53 Not continuous 55 Continuous 57 Not continuous 59 Continuous 61 Continuous for all real numbers 63 Continuous for all real numbers 65 Continuous for all real numbers kp 67 Continuous for all real numbers except x = , where k is an odd integer 69 Continuous for all real numbers except x = - and x = 2 71 Continuous for all positive real numbers except x = 73 Discontinuous at x = - and x = 1; 1 lim R1x2 = : hole at a1, b xS1 2 lim -R1x2 = - q ; lim + R1x2 = q ; 75 Discontinuous at x = - and x = 1; 1 lim R1x2 = : hole at a - 1, b x S -1 2 lim-R1x2 = - q ; lim+ R1x2 = q ; y 5 x y؍0 x S -1 x S -1 vertical asymptote at x = - 79 x = - 3: asymptote; x = 2: hole 85 20 −20 20 −3 91 Vertical: x = 4; horizontal: y = −20 92 60 y؍1 x xS1 vertical asymptote at x = x ؍؊1 77 x = - 2 : asymptote; x = 1: hole 83 −3 xS1 y 93 ln ¢ x 5y z4 ≤ 94 C x؍1 81 x = - 2 : asymptote; x = - 1: hole 87 −3 −2 1 23 5S -3 -2 www.ebookslides.com AN101 ANSWERS Section 13.5 13.4 Assess Your Understanding (page 948) tangent line derivative velocity mtan = True True True 11 mtan = - 13 mtan = 12 y 10 (1, 8) f (x) ؍3x f (x) ؍x ؉ (؊1, 3) (2, 12) f (x) ؍2x ؉ x 17 mtan = - y 18 10 y ؍13x ؊ 16 (؊1, 6) (2, 10) f(x) ؍x ؊ 2x ؉ x f (x) ؍x ؉ x x 52 (2, 6), ( - 1, 3) 53 10 x y ؍5x ؊ x 19 mtan = 13 y (1, 3) y ؍12x ؊ 12 x x 51 Vertex: (1, 3); focus: ¢ 1, ≤ y y 15 y ؍؊2x ؉ f(x) ؍3x ؉ y ؍؊4x ؉ 15 mtan = y 10 21 - 23 25 27 29 31 33 60 35 - 0.8587776956 37 1.389623659 39 2.362110222 41 3.643914112 43 18p ft 3/ft 45 16p ft 3/ft 47 (a) sec (b) 64 ft/sec (c) - 32t + 962 ft/sec (d) 32 ft/sec (e) sec (f) 144 ft (g) - 96 ft/sec 49 (a) - 23 ft/sec (b) - 21 ft/sec (c) - 18 ft/sec (d) s 1t2 = - 2.631t - 10.269t + 999.933 (e) Approximately - 15.531 ft/sec 54 23.66 sq units 13.5 Assess Your Understanding (page 955) b La (a) f 1x2 dx b La f 1x2 dx 56 11 (a) y 24 13 (a) y 10 15 (a) y 20 y 90 x (b) 36 (d) 45 17 (a) (c) 72 (e) 63 (f) 54 (b) 18 (c) 63 45 (d) (e) 4 19 (a) y 1.5 x x x 51 (b) 22 (c) 27 (f) (d) 21 (a) y 20 L0 (b) 36 1x2 + 22 dx (e) 88 y 1.2 (d) L0 x3 dx (e) 64 23 (a) Area under the graph of f 1x2 = 3x + from to (b) y P x x (c) 49 16 x (b) 25 12 (c) (b) 11.475 4609 2520 (c) 15.197 (b) 1.896 p (c) 1.974 x x (d) e dx (e) 19.718 (d) sin x dx (e) (c) 28 L-1 L0 dx (e) 1.609 (d) x L1 27 (a) Area under the graph of f 1x2 = sin x 29 (a) Area under the graph of f 1x2 = e x from 25 (a) Area under the graph of f 1x2 = x2 - p to from to from to (b) y (b) y (b) y 30 1.5 P x x x (c) 36 (c) 31 Using left endpoints: n = 2: + 0.5 = 0.5; n = 4: + 0.125 + 0.25 + 0.375 = 0.75; 10 10 + 0.182 = 0.9; n = 100: + 0.0002 + 0.0004 + 0.0006 + g + 0.0198 100 = 10 + 0.01982 = 0.99; n = 10: + 0.02 + 0.04 + 0.06 + g + 0.18 = (c) 6.389 Using right endpoints: n = 2: 0.5 + = 1.5; n = 4: 0.125 + 0.25 + 0.375 + 0.5 = 1.25; 10 10.02 + 0.202 = 1.1; n = 10: 0.02 + 0.04 + 0.06 + g + 0.20 = n = 100: 0.0002 + 0.0004 + 0.0006 + g + 0.02 100 10.0002 + 0.022 = 1.01 = www.ebookslides.com AN102 y 33 ANSWERS Section 13.5 34 J f(x) log2x 22 19 43 22 50 R 35 4x + 2h + 36 1 x - x + (x + 2)2 x Review Exercises (page 957) 1 3 10 11 12 Continuous 13 Not continuous 14 Continuous 15 Not continuous 2 16 5x͉ - … x or x or x … 6 17 All real numbers 18 1, 19 20 f - 62 = 2; f - 42 = 21 22 - 23 - q 24 q 25 Does not exist 26 No 27 Yes - 343 4 25 27 - 28 R is discontinuous at x = - and x = 1 lim R1x2 = - : hole at a - 4, - b x S -4 8 lim-R1x2 = - q ; lim+ R1x2 = q : 29 Undefined at x = and x = 9; R has a hole at x = and a vertical asymptote at x = y 5 x y؍0 xS4 xS4 The graph of R has a vertical asymptote at x = x؍4 31 mtan = 30 mtan = 12 y 21 f (x) ؍2x ؉ 8x y 32 mtan = 16 y 21 f (x) ؍x ؉ 2x ؊ (2, 12) (1, 10) x 10 x (؊1, ؊4) y ؍12x ؊ x y ؍16x ؊ 20 f (x) ؍x ؉ x y ؍؊4 33 - 24 34 - 35 36 - 158 37 0.6662517653 38 (a) sec (b) sec (c) 64 ft/sec (d) - 32t + 962 ft/sec (e) 32 ft/sec (f) At t = sec (g) - 96 ft/sec (h) - 128 ft/sec 39 (a) $61.29/watch (b) $71.31/watch (c) $81.40/watch (d) R1x2 = - 0.25x2 + 100.01x - 1.24 (e) Approximately $87.51/watch 40 (a) 41 (a) y 14 42 (a) y 43 (a) Area under the graph of f 1x2 = - x2 from - to (b) y y 10 x x x (b) 24 (d) 26 (c) 32 (e) 30 (f) 28 77 (c) (b) 10 (d) L-1 49 ≈ 1.36 (c) 1.02 36 (d) dx (e) 0.75 L1 x2 (b) 14 - x2 dx 44 (a) Area under the graph of f 1x2 = e x from - to (b) (e) x (c) (c) 2.35 y 2.5 x Chapter Test (page 959) - 135 - - 10 11 Limit exists; 3 12 (a) Yes (b) No; lim f 1x2 ≠ f 112 (c) No; lim- f 1x2 ≠ f 132 (d) Yes 13 x = - 7: asymptote; x = 2: hole - xS1 xS3 14 (a) (b) y = 5x - 19 (c) f(x) ؍4x ؊ 11x ؊ y 15 (a) 16 y 5 x 30 y ؍5x ؊ 19 x (2, ؊9) (b) 13.359 (c) 4p ≈ 12.566 L1 - x2 + 5x + 32 dx 17 35 ft/sec 80 www.ebookslides.com AN103 ANSWERS Section A.4 APPENDIX A Review A.1 Assess Your Understanding (page A10) variable origin strict base; exponent or power True True False False 51, 2, 3, 4, 5, 6, 7, 8, 11 54 13 51, 3, 4, 6 15 50, 2, 6, 7, 17 50, 1, 2, 3, 5, 6, 7, 8, 19 50, 1, 2, 3, 5, 6, 7, 8, 21 – 2.5 –1 31 23 25 0.25 27 51 - 28 29 = 53 31 55 77 5x͉ x ≠ - 105 - 57 79 0°C 107 131 C = pd 33 x 109 59 35 x 61 81 25°C 111 37 x … 63 22 83 16 113 15 115 39 41 67 x = 65 85 16 117 10; 87 69 x = 89 91 119 81 47 49 95 64x6 121 304,006.671 75 5x͉ x ≠ 73 x = 0, x = 1, x = - 71 None 93 23 x 135 V = pr 137 V = x3 139 (a) $6000 147 No; is larger; 0.000333 c 149 No (b) No 45 –1 133 A = 145 (a) Yes 43 –2 97 123 0.004 x y 99 x y 101 - 125 481.890 141 ͉ x - 4͉ Ú (b) $8000 8x3z 9y 127 0.000 103 16x2 9y2 129 A = lw 143 (a) … (b) A.2 Assess Your Understanding (page A19) bh C = 2pr similar True True False True True 10 False 11 13 13 26 15 25 17 Right triangle; 19 Not a right triangle 21 Right triangle; 25 23 Not a right triangle 25 in2 27 in2 29 A = 25p m2; C = 10p m 256 31 V = 224 ft 3; S = 232 ft 33 V = p cm3; S = 64p cm2 35 V = 648p in3; S = 306p in2 37 p square units 39 2p square units 41 x = units; A = 90°; B = 60°; C = 30° 43 x = 67.5 units; A = 60°; B = 95°; C = 25° 45 About 16.8 ft 47 64 ft right; hypotenuse A = 49 24 + 2p ≈ 30.28 ft 2; 16 + 2p ≈ 22.28 ft 51 160 paces 53 About 5.477 mi 55 From 100 ft: 12.2 mi; From 150 ft: 15.0 mi A.3 Assess Your Understanding (page A30) 4; x4 - 16 x3 - False True False Monomial; variable: x; coefficient: 2; degree: Not a monomial; the exponent of the variable is not a nonnegative integer 11 Monomial; variables: x, y; coefficient: - 2; degree: 13 Not a monomial; the exponent of one of the variables is not a nonnegative integer 15 Not a monomial; it has more than one term 17 Yes; 19 Yes; 21 No; the exponent of the variable of one of the terms is not a nonnegative integer 23 Yes; 25 No; the polynomial of the denominator has a degree greater than 27 x2 + 7x + 29 x - 4x + 9x + 31 6x + 5x + 3x + x 33 7x2 - x - 35 - 2x3 + 18x2 - 18 37 2x2 - 4x + 39 15y2 - 27y + 30 2 2 41 x + x - 4x 43 - 8x - 10x 45 x + 3x - 2x - 47 x + 6x + 49 2x + 9x + 10 51 x - 2x - 53 x2 - 5x + 55 2x2 - x - 57 - 2x2 + 11x - 12 59 2x2 + 8x + 61 x2 - xy - 2y2 63 - 6x2 - 13xy - 6y2 65 x2 - 49 67 4x2 - 69 x2 + 8x + 16 71 x2 - 8x + 16 73 9x2 - 16 75 4x2 - 12x + 77 x2 - y2 79 9x2 - y2 81 x2 + 2xy + y2 83 x2 - 4xy + 4y2 85 x3 - 6x2 + 12x - 87 8x3 + 12x2 + 6x + 89 4x2 - 11x + 23; remainder - 45 91 4x - 3; remainder x + 1 2 2 99 - 4x2 - 3x - 3; remainder - 93 5x - 13; remainder x + 27 95 2x ; remainder - x + x + 97 x - 2x + ; remainder x + 2 101 x2 - x - 1; remainder 2x + 103 x2 + ax + a2; remainder A.4 Assess Your Understanding (page A39) 3x(x - 2)(x + 2) 15 (x + 1)(x - 1) 29 (2x + 1) 2 prime 79 (x + 1)(x + 10) 81 (x - 7)(x - 3) 95 2(3x + 1)(x + 1) 123 2(3x + 4)(9x + 13) 35 (x + 3)(x - 3x + 9) 69 25; (x + 5)2 83 4(x2 - 2x + 8) 47 (x - 8)(x + 1) 71 9; ( y - 3)2 85 Prime 37 (2x + 3)(4x - 6x + 9) 49 (x + 8)(x - 1) 27 (x - 5)2 39 (x + 2)(x + 3) 51 (x + 2)(2x + 3) 61 (x + 2)(3x - 4) 63 (x - 2)(3x + 4) 1 ; ax - b 75 (x + 6)(x - 6) 77 2(1 + 2x)(1 - 2x) 16 87 - (x - 5)(x + 3) 89 3(x + 2)(x - 6) 91 y2( y + 5)( y + 6) 73 99 (x - 1)2(x2 + x + 1)2 119 (x - 1)(x + 1)(x + 2) 127 5(x + 3)(x - 2)2(x + 1) 25 (x + 2) 23 (x + 1) 13 3xy(x - 2y + 4) 2 109 - (3x - 1)(3x + 1)(x2 + 1) 117 (x + 5)(3x + 11) 11 2x(x - 1) 59 (z + 1)(2z + 3) 97 (x - 3)(x + 3)(x + 9) 125 2x(3x + 5) x(x2 + x + 1) 21 (5x + 2)(5x - 2) 57 (3x + 1)(x + 1) 107 (2y - 5)(2y - 3) 115 13x - 52 19x - 3x + 72 a(x2 + 1) 45 (x - 8)(x - 2) 55 (2x + 3)(3x + 2) 67 (x + 4)(3x - 2) 105 - (4x - 5)(4x + 1) 19 (x + 4)(x - 4) 33 (x - 3)(x + 3x + 9) 65 (x + 4)(3x + 2) 93 (2x + 3) 3(x + 2) 43 (x + 5)(x + 2) 53 (x - 2)(2x + 1) False 17 (2x + 1)(2x - 1) 31 (4x + 1) 41 (x + 6)(x + 1) True 101 x5(x - 1)(x + 1) 111 (x + 3)(x - 6) 113 (x + 2)(x - 3) 121 (x - 1)(x + 1)(x - x + 1) 129 3(4x - 3)(4x - 1) 131 6(3x - 5)(2x + 1)2(5x - 4) 133 The possibilities are (x { 1)(x { 4) = x { 5x + or (x { 2)(x { 2) = x { 4x + 4, none of which equals x2 + 2 103 (4x + 3)2 www.ebookslides.com AN104 ANSWERS Section A.5 A.5 Assess Your Understanding (page A44) True True x2 + x + 4; remainder 12 11 4x5 + 4x4 + x3 + x2 + 2x + 2; remainder 19 Yes 21 Yes 23 No 25 Yes 27 - quotient; divisor; remainder - 3) - x4 - 3x3 + 5x2 - 15x + 46; remainder - 138 15 x4 + x3 + x2 + x + 1; remainder 17 No A.6 Assess Your Understanding (page A53) lowest terms least common multiple True False 19 2x(x2 + 4x + 16) x + 21 3x 23 x - x + 25 x - 4x (x - 2)(x - 3) (x - 2)(x + 2) x + 3x - x + - x 37 39 41 43 2x - x - 2x - x - 2(x2 - 2) 51 53 (x - 2)(x + 2)(x + 1) 55 x(x - 1)(x + 1) x(x - 2)(x + 2) 5x (x - 6)(x - 1)(x + 4) 71 -1 x(x + h) 87 73 x + x - (x + 1)(x - 1) (x + 1) 63 89 75 2(2x2 + 5x - 2) 65 (x - 2)(x + 2)(x + 3) (x - 1)(x + 1) 77 2x(2x + 1) x(3x + 2) (3x + 1) (x - 1) (x + 1) 79 93 f = y + 4x 47 (x - 1)(x + 2) 67 13 2( y + 1) (x - 4)2 57 x3(2x - 1)2 (x - 2)(x + 1) (x + 1) 29 - 11 2(x + 5) 45 2(5x - 1) 4x 2x - 5x + (x + 3)(3x - 1) 91 - 5(x - 1) 27 35 61 x 3x2 + 11x + 32; remainder 99 13 0.1x2 - 0.11x + 0.321; remainder - 0.3531 31 x + x - 15 - (x + 7) (x + 3)2 33 (x - 3)2 3x2 - 2x - (x + 1)(x - 1) 17 5x(x - 2) (x - 4)(x + 3) (x - 1)(2x + 1) 49 - (11x + 2) (x + 2)(x - 2) 59 x(x - 1)2(x + 1)(x2 + x + 1) - x2 + 3x + 13 (x - 2)(x + 1)(x + 4) - 2x(x2 - 2) 81 (x + 2)(x2 - x - 3) R1 # R2 ; m (n - 1)(R1 + R2) 15 69 -1 x - x3 - 2x2 + 4x + x2(x + 1)(x - 1) 83 3x - 2x + 85 19 (3x - 5)2 A.7 Assess Your Understanding (page A60) index True 25 12 23 43 22 65 27 22 32 81 27 22 45 - 67 x7>12 + x 2(1 + x) 215 3>2 83 cube root 29 22 47 15 93 (x2 + 4)1>3(11x2 + 12) 107 (a) 15,660.4 gal 49 23 71 x2>3y - x 3>2 85 73 8x5>4 y x (x - 1) 1>2 87 15 x3y2 17 x2y 19 1x 52 2x + h - 2x2 + xh h 55 x(3x2 + 2) 22x + 51 53 3x + (1 + x) 77 1>2 - 3x2 1x(1 + x2)2 95 (3x + 5)1>3(2x + 3)1>2(17x + 27) (b) 390.7 gal 13 - 2x x 37 (2x - 1) 2x 75 3>4 11 22 35 x - 1x + 25 - 19 41 - 33 - 2 31 23 + 22 23 69 xy2 (x + 4) False 109 22p ≈ 8.89 sec 111 97 (x2 + 1)1>2 79 39 (2x - 15) 22x 2x1>2 99 1.41 41 - (x + 5y) 2xy 59 64 61 27 63 27 22 32 10 1x - 52 14x + 32 89 (5x + 2)(x + 1)1>2 3(x + 2) 57 - 3 23 15 21 6x 1x 91 2x1>2(3x - 4)(x + 1) 101 1.59 103 4.89 105 2.15 p 23 ≈ 0.91 sec A.8 Assess Your Understanding (page A70) 5 equivalent equations identity extraneous True {4} 11 e f 13 - 15 - 18 17 - 19 - 16 21 50.5 23 {2} 25 {2} 27 {3} 29 {0, 9} 31 - 3, 0, 33 {21} 35 {6} 37 ∅ or { } 39 - 5, 0, 41 - 1, 43 e - 2, , f 45 {1} 47 No real solution 49 - 13 51 {3} 53 {8} 55 - 1, 57 {1, 5} 59 {5} 61 {2} 63 e - 11 f 65 - 6 67 R = R1R2 R1 + R2 69 R = mv2 F 71 r = S - a S 73 The distance is approximately 229.94 ft 75 Approx 221 ft 79 The apparent solution - is extraneous 81 When multiplying both sides by x + 3, we are actually multiplying both sides by when x = - This violates the Multiplication Property of Equality A.9 Assess Your Understanding (page A78) mathematical modeling interest uniform motion False True 100 - x A = pr 2; r = radius, A = area A = s2; A = area, s = length of a side 11 F = ma; F = force, m = mass, a = acceleration 13 W = Fd; W = work, F = force, d = distance 15 C = 150x; C = total variable cost, x = number of dishwashers 17 Invest $31,250 in bonds and $18,750 in CDs 19 $11,600 was loaned out at 8% 21 Mix 75 lb of Earl Grey tea with 25 lb of Orange Pekoe tea 23 Mix 160 lb of cashews with the almonds 25 The speed of the current is 2.286 mi/hr 27 The speed of the current is mi/hr 29 Karen walked at 4.05 ft/sec 31 A doubles tennis court is 78 feet long and 36 feet wide 33 Working together, it takes 12 35 (a) The dimensions are 10 ft by ft (b) The area is 50 sq ft (c) The dimensions will be 7.5 ft by 7.5 ft (d) The area will be 56.25 sq ft 2 37 The defensive back catches up to the tight end at the tight end’s 45-yd line 39 Add gal of water 41 Evaporate 10 oz of water 3 43 40 g of 12-karat gold should be mixed with 20 g of pure gold 45 Mike passes Dan mile from the start, from the time Mike started to race 47 The latest the auxiliary pump can be started is 9:45 am 49 The tub will fill in hr 51 Run: 12 miles; bicycle: 75 miles 53 Bolt would beat Burke by 19.75 m 55 Set the original price at $40 At 50% off, there will be no profit 59 The tail wind was 91.47 knots www.ebookslides.com AN105 ANSWERS Section A.11 A.10 Assess Your Understanding (page A86) negative closed interval 13 32, q 2; x Ú (c) 12 - 15 0, 32; … x (d) - 6 17 (a) 6 21 (a) 2x + 23 [0, 4] True (b) 2x - - 4 31 … x … ؊3 41 43 Ú 45 53 {x͉ x 4} or - q , 42 47 … 49 77 5x͉ x - or - q ,- 52 10 10 r or a , q b 3 37 x - ؊3 57 {x͉ x 3} or 13, q 2 59 {x͉ x Ú 2} or [2, q 2 65 {x͉ x - 20} or ( - q , - 20) 67 b x ` x Ú 4 r or c , q b 3 ؊20 2 … x … r or c , d 3 79 5x͉ x Ú - or - 1, q ؊1 73 b x ` ؊ 81 b x ` 11 11 x r or a - , b 2 2 11 75 5x͉ - 6 x 6 or - 6, 02 ؊6 5 … x r or c , b 4 87 5x͉ x or 13, q 10 ؊4 2 f or a - q , d 3 ؊5 29 - q , - 42 71 b x ` (b) - - 51 Ú ؊7 69 5x͉ … x … or 3, 19 (a) 7 (d) - 4x - - 27 4, q 11 [0, 2]; … x … 10 True 35 x Ú 55 {x͉ x Ú - 1} or [ - 1, q 63 e x͉ x … False (d) - - 10 (c) 6x + 6 −1 61 {x͉ x - 7} or - 7, q 85 b x ` x (c) 15 ؊2 True 33 - x - 2 True (b) - 25 [4, 6) 39 Multiplication Property 1 83 b x ` x - r or a - q , - b 2 89 5x͉ x Ú - ؊ 91 21 Age 30 93 (a) Male Ú 77.3 years (b) Female Ú 81.6 years (c) A female can expect to live 4.3 years longer 95 The agent’s commission ranges from $45,000 to $95,000, inclusive As a percent of selling price, the commission ranges from 5% to 8.6%, inclusive 97 The amount withheld varies from $111.20 to $161.20, inclusive 99 The usage varies from 650 kW # hr to 2500 kW # hr, inclusive 101 The dealer’s cost varies from $15,254.24 to $16,071.43, inclusive 103 (a) You need at least a 74 on the fifth test (b) You need at least a 77 on the fifth test a + b a + b - 2a b - a a + b a + b 2b - a - b b - a a + b - a = = 0; therefore, a b = = 0; therefore, b 105 2 2 2 2 2 2 107 1ab2 - a = ab - a = a 1b - a2 0; thus 1ab2 a and 1ab a b2 - 2ab2 = b2 - ab = b 1b - a2 0; thus b2 2ab2 and b 2ab a 1b - a2 b 1b - a2 ab - a2 2ab b2 - ab 2ab - a = = 0; thus h a b - h = b = = 0; thus h b a + b a + b a + b a + b a + b a + b a - b a b 1 1 111 Since a b, then a - b and So 0, or Therefore, And because b ab ab ab b a b a b 113 x2 + Ú for all real numbers x 109 h - a = A.11 Assess Your Understanding (page A94) real; imaginary; imaginary unit - 1; - i; False True True + 5i 11 - + 6i 13 - - 11i 15 - 18i 23 19 10 - 5i 21 37 23 + i 25 - 2i 27 - i 29 - + i 31 2i 33 - i 35 i 37 - 39 - 10i 41 - + 2i 5 2 2 47 2i 49 5i 51 5i 53 55 25 57 + 3i 59 z + z = 1a + bi2 + 1a - bi2 = 2a; z - z = 1a + bi2 - 1a - bi2 = 2bi 61 z + w = 1a + bi2 + 1c + di2 = 1a + c2 + 1b + d2i = 1a + c2 - 1b + d2i = 1a - bi2 + 1c - di2 = z + w 67 2a # 2b = 2ab only when 2a and 2b are real numbers 17 + 4i 43 45 AN106 www.ebookslides.com ANSWERS Section B.1 APPENDIX B Graphing Utilities B.1 Exercises (page B2) ( - 1, 4); II (3, 1); I Xmin = - 6, Xmax = 6, Xscl = 2, Ymin = - 4, Ymax = 4, Yscl = Xmin = - 6, Xmax = 6, Xscl = 2, Ymin = - 1, Ymax = 3, Yscl = Xmin = 3, Xmax = 9, Xscl = 1, Ymin = 2, Ymax = 10, Yscl = 11 Xmin = - 11, Xmax = 5, Xscl = 1, Ymin = - 3, Ymax = 6, Yscl = 13 Xmin = - 30, Xmax = 50, Xscl = 10, Ymin = - 90, Ymax = 50, Yscl = 10 15 Xmin = - 10, Xmax = 110, Xscl = 10, Ymin = - 10, Ymax = 160, Yscl = 10 B.2 Exercises (page B4) (a) (b) ؊5 ؊10 ؊5 ؊5 ؊10 ؊5 ؊10 ؊10 ؊8 ؊10 ؊4 19 21 23 25 27 29 31 B.3 Exercises (page B6) - 1.71 - 0.28 3.00 4.50 11 1.00, 23.00 B.5 Exercises (page B8) Yes Yes No Yes 10 ؊8 17 - 3.41 10 ؊8 (b) ؊5 10 ؊4 ؊4 15 (a) 10 ؊8 (b) ؊8 (b) ؊10 ؊5 10 ؊4 13 (a) ؊4 11 (a) 10 ؊8 (b) ؊8 (b) ؊10 ؊5 10 ؊4 (a) (a) ؊10 ؊4 ؊8 (b) (b) ؊5 10 ؊4 (a) (a) Answers may vary A possible answer is Ymin = 4, Ymax = 12, and Yscl = www.ebookslides.com Photo Credits 'SPOU.BUUFS Page 25, E+/Getty Images $IBQUFS' Pages 33 and 73, Andy Dean Photography/Shutterstock; Page 44, Barrett & MacKay/Glow Images; Page 50, Dept of Energy (DOE) Digital Photo Archive; Page 64, Tetra Images/Alamy; Page 71, Courtesy of Caesars Entertainment $IBQUFS Pages 74 and 148, Stephen Coburn/Shutterstock; Page 87, NASA; Page 95, Superstock; Page 132, Dreamstime $IBQUFS Pages 150 and 221, Luis Louro/Shutterstock; Page 204, Geno EJ Sajko/ Shutterstock $IBQUFS Pages 223 and 303, Aleksey Stemmer/Shutterstock; Page 289, Corbis $IBQUFS Pages 305 and 406, Fotolia; Page 363, Getty Images; Pages 370, 376, and 390, Thinkstock; Page 371, Superstock $IBQUFS Pages 407 and 493, Nova for Windows/NLSA; Page 419, Ryan McVay/ Thinkstock; Page 466, Srdjan Draskovic/Dreamstime $IBQUFS Pages 495 and 562, Sebastian Kaulitzki/iStockphoto $IBQUFS Pages 563 and 614, Jennifer Thermes/Getty Images; Page 575, Digital Vision/ eStock Photo; Page 598, Alexandre Fagundes De Fagundes/Dreamstime; Page 600, iStockphoto/Thinkstock $IBQUFS Pages 615 and 689, Fotolia; Page 637, North Wind Picture Archives/Alamy; Page 646, Science & Society Picture Library/Getty Images; Page 658, Hulton Archive/ Getty Images $IBQUFS Pages 691 and 753, NASA; Page 708, Thomas Barrat/Shutterstock $IBQUFS Pages 754 and 850, Wavebreakmedia/Shutterstock; Page 808, Library of Congress and Photographs Division [LC-USZ62-46864] $IBQUFS Pages 851 and 893, Denis Cristo/Shutterstock; Pages 875 and 875, Pearson Education, Inc $IBQUFS Pages 894 and 922, NBCUniversal/Getty Images; Page 915, iStockphoto/ Thinkstock $IBQUFS Pages 923 and 960, Rafael Macia/Photo Researchers, Inc "QQFOEJY" Page A14, Feraru Nicolae/Shutterstock C1 www.ebookslides.com www.ebookslides.com Subject Index Δ (change in), 52, 104 ± (plus or minus), 67 Abel, Niels, 255, 888 Abscissa, 34 Absolute maximum and absolute minimum of function, 102–3 Absolute value, 641, A5 Absolute value equations, 210–11 Absolute value function, 111–12, 114 Absolute value inequalities, 211–13 Acute angles, 564–66 Addition See also Sum of complex numbers, A90 of polynomials, A24–A25 of rational expressions, A47–A48 least common multiple (LCM) method for, A48–A50 triangular, 888 of vectors, 649, 652–53 in space, 672 Addition principle of counting, 896–97 Addition property of inequalities, A83 Additive inverse, A47 Adjacency matrix, 811 Ahmes (Egyptian scribe), 875 Airfoil, 615 Airplane wings, 615 Algebra essentials, A1–A13 distance on the real number line, A5–A6 domain of variable, A7 evaluating algebraic expressions, A6–A7 evaluating exponents, A10 graphing inequalities, A4–A5 Laws of Exponents, A7–A9 real number line, A4 sets, A1–A4 to simplify trigonometric expressions, 525 square roots, A9–A10 Algebraic functions, 305 Algebraic vector, 651–52 Algorithm, 245n Alpha particles, 723 Altitude of triangle, A15 Ambiguous case, 578–79 Amount of annuity, 874–75 Amplitude of simple harmonic motion, 600 of sinusoidal functions, 456–58, 460, 476–79 Analytic trigonometry, 495–62 algebra to simplify trigonometric expressions, 525 Double-angle Formulas, 543–53 to establish identities, 544–47 to find exact values, 544 Half-angle Formulas, 547–50 to find exact values, 547–50 for tangent, 549–50 Product-to-Sum Formulas, 553–55 Sum and Difference Formulas, 531–43 for cosines, 531–32 defined, 531 to establish identities, 533–36 to find exact values, 532, 534–35 involving inverse trigonometric function, 537 for sines, 534 for tangents, 536 Sum-to-Product Formulas, 555 trigonometric equations, 514–23 calculator for solving, 517 graphing utility to solve, 519 identities to solve, 518–19 involving single trigonometric function, 514–17 linear, 515–16 linear in sine and cosine, 538–39 quadratic in from, 517–18 solutions of, defined, 514 trigonometric identities, 523–30 basic, 524 establishing, 525–28, 533–36, 544–47 Even-Odd, 524 Pythagorean, 524 Quotient, 524 Reciprocal, 524 Angle(s), 408–21 acute, 564–66 central, 411 converting between decimals and degrees, minutes, seconds forms for, 410–11 of depression, 568, 569 direction, 655, 675–77 drawing, 409–10 of elevation, 568 elongation, 586 of incidence, 522 of inclination, 438 initial side of, 408, 409 negative, 408 optical (scanning), 552 positive, 408 quadrantal, 409, 425–27 of refraction, 522 of repose, 513 right, 409, A13 in standard position, 408 straight, 409 terminal side of, 408, 409 trigonometric functions of, 424–27 between vectors, 663–64, 674–75 viewing, 507 Angle–angle case of similar triangle, A16, A17 Angle–side–angle case of congruent triangle, A16 Angular speed, 416 Annuity(ies), 874–75 amount of, 874–75 defined, 874 formula for, 874 ordinary, 874–75 Aphelion, 710, 737, 753 Apollonius of Perga, 691 Applied (word) problems, A72–A80 constant rate job problems, A77 interest problems, A73–A74 mixture problems, A74–A75 steps for solving, A73 translating verbal descriptions into mathematical expressions, A72 uniform motion problems, A75–A77 Approaches infinity, 265–66 Approximate decimals, A3 Approximating area, 951–54 Araybhatta the Elder, 433 Arc length, 411–12 Area definition of, 954 formulas for, A15 under graph of function, 951–54 of parallelogram, 683 of triangle, 593–99, A15 SAS triangles, 594–95 SSS triangles, 595–96 Argument of complex number, 641 of function, 79 Arithmetic calculator, A10 Arithmetic mean, A89 Arithmetic progression See Arithmetic sequences Arithmetic sequences, 862–68 common difference in, 862 defined, 862 determining, 862–63 formula for, 863–64 nth term of, 863 recursive formula for, 863–64 sum of, 864–66 Ars Conjectandi (Bernoulli), 916 Ars Magna (Cardano), 255 ASA triangles, 578 Associative property of matrix addition, 797 of matrix multiplication, 802 of vector addition, 649 Asymptote(s), 266–72, 716–18 horizontal (oblique), 267, 269–72, 321 vertical, 267, 268–69, 321 Atomic systems, 808 Augmented matrix, 770–71, 772 in row echelon form, B9–B10 Average cost function, 92 Average rate of change, 52, 329–30 of function, 104–6 defined, 104 finding, 104–5 secant line and, 105–6 limit of, 934 of linear functions, 151–54 Axis/axes of complex plane, 640–41 of cone, 692 coordinate, 34 of ellipse, 702, 732 of hyperbola conjugate, 712 transverse, 712, 732 polar, 616–17 of quadratic function, 183–84 rotation of, 725–27 analyzing equation using, 727–29 formulas for, 726 identifying conic without, 729–30 of symmetry of parabola, 182, 693 of quadratic function, 183–84 Azimuth, 571n Babylonians, ancient, 875 Back substitution, 758 Barry, Rick, 95 Base of exponent, A8 Basic trigonometric identities, 524 Bearing (direction), 571 Bernoulli, Jakob, 637, 916 Bernoulli, Johann, 746 Bessel, Friedrich, 573 Bessel’s Star, 573 Best fit line of, 164–65 polynomial function of, 238–39 quadratic functions of, 201–2 Beta of stock, 150, 221–22 Bezout, Étienne, 826 Binomial(s), A23 cubing, A27 squares of (perfect squares), A26 Binomial coefficient, 886 Binomial Theorem, 883–89 n to evaluate a b , 884–85 j expanding a binomial, 886–88 historical feature on, 888 I1 ... 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