LIVRO DE CÁLCULO VOL. 2

 

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livro de cálculo

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geotuetry formulas rr.tr para e ogranl [rrr]l {illff ]rr.r tr arg e ljcrsht r=rr.r lbr trapezo d rrllu -nrnrllishl flr.utrllrrn.c xr.l.n!rh sector r l 11 u+bth grl i r.j -6 tl n rao a 1 .t rr c=lzr nsi r ar cy fder rrght crrcl ar cone any cylnder or pr sm wth paralle bases sphere l /ffr ffir l =l;r s 1;r l algebra formulas the quadratic formula i cqurti,rr 1 ail l =0arc /r ii,r -tdr the binomial formula r i rr rr r ir table of integrals 1!r lt1_1 l lr ,rr .1,n11r l;y basic functions l ,t e i l =ttl n i1 a 1 lr,r tt u r r r c .u r t inr r .l -u t lr a co r r rn sec t n r u 1 in r rn =tn rrlnl;r 1,i +c t scc r r.i r ,r r l .r a.o .r-rrr 1 rrr c cr-cot :tn ranlrr +c .u r or r tn t r f r r rtanlrr ln r l rrr c l f 1 ,

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rational functions containing powers of a bu ln the denominator i l j u t utdu t unu tfait1 lt+t1u j ta+bu l tndrhd -t t,lu i t a o li 4 ola bu a no l l -lhu i t ottllu+h/ll+c bu i c -c 1 6 7 jr rl t/llu ou 1 t i jta.b ht lz,r i bu a buri i i ln u +a i du i ut.t+but a a+hu i t h ln u+bultti du j,2,attu i dt i t i h ala tu i a ln a bu jiii udu iiuii tl/1 rattonal functtons contatntng a2 u2 tn the denomtnator a 0 68 i rj lan j a i u 6r i du 1 i tn j a u ta r c u+al u-a l+c 70 i f nu i u nl c j u a 2u u 11 hu -1 if ln lu h -l tu -ln 2 ,t o t4 o j ran c integrals of a2 u2 a2-u2,r u2 a2 and their reciprocals {a lha i ji r r o a ljt +a 7tn t o u t a i r^u-ju 1 1+c a2 7 7t -r re ;a i l powers of u multiplying or dividing l d l a a,dtt -ol 7 7 u2 or its reciprocal ib r.g-a x0 i .luua u s5 l g td irt nf f 1+c i -7 e 1 8r 82 9 -c 2 a2 or 5 1 in i l j u a u i rf l t ju j u2 jal u2 c i uzdu i _j la u t dr o t 1 -i,in i o j,tl u1l li i powers of u multiplying or dividing titin reciprocals i urlf a ttu !s1 n2ft2 5 i u lf -z d !a o t c ,1,t _t 1,ffi uju j i du tl7 l i uj u a r 2 i sec u +c l u1 -e itu rrf ou +c ju i di 1 fr u23 integrals contatntng a2 u23 2,1u2 2,lu2 u23 2 r>o i j a u du |i j gt lotlttt ee ju a a u2 u ro0 i e 21/2 l a2 c ju +a2 r:i i e irz si1,if a r.r i ;t c a11/2 ,tu itz s nlf a rn i 1 +c d u23t2,ju f,12 1 s t.r j c $ri 1 +c

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reciprocals of basic functions j 1+sin r ld corx+csc c t1 ,n l d i r lrlcos rn 1 du n ral,tt+c 2r powers of trigonometric functions ft a tin sec ,t t1 r,du .l tu+tn,rr co.a c -l dri corr c l j lf sec 24 du-u tanu e-u+e j 1+csc ls du:u tnt u c .l t+e ze 27 28 ,u i 1,a j i sin2x c zz ,d i 2 c j lcouau cotu-u+c u+c t .r5 r la a l a products of trigonometric functions t t ,t zo a j f 1 0 .n 1 lan2uau:tanu-u+c /coi ridr n-t .o iror j j sec uau 1 ,a .o,u-c lsec2udu=ral uou n i 1 y a csc 2acor fi a co n n co ifl n 2n.nt pn-n1 aco r !ucos ud n n n.t ,au fi a i .r8 i sinzasrnza/a j t 39 i co nu,o nudu .l intm i ht n ht r 2n i n 2n -tc ntn t nt n j ltm n t tln 40 t j n a j 41 srn l rco t n+d n+n i n o products of trigonometric and exponential functions 42 i e nhu iu ,1 tinbu hco bu a d .1 i par c n i a .l co,un at .cl a.o bl a tnbut c a d powers of u multiplying or dividing easic functions 44 i i usinudu sinu ucosu i c 51 .t i ue au e ru tt+c 1 d cos xsin 46 u2 sn d :2,sin 2 47 +c a cos t+c 53 5,1 tn l .t u o .t n iu o,o n t na ina 1na j j 2!cosll x2 2sin +c 1 1 s t $r i d u cosu+n u lcosudu i 55 56 so j rn-ttu t tl ia du att lna ia d i -t tj n :lnlln4 i e d t i i e dd u cosudu:u sin f 1 s;nuau +c n,a ffi,r inot a v t lnal in.ra r,t polynomials multiplying basic functions 57 58 5g t ro lsisns artemate p us;nau p ucosau ln t o a no aw }o rr p cosau lsigns altemate in pairs after first term [sisns a]temate in pairs .l 1

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,c rers 0f l,4lllt pl l lc cr dlvidli lg n bl 0ll iti il:cipqccal 1 -n ,t+ht t a jd+fu d atlu r ln rr llr lr i i iic 2 -r 0r s eec procal 1 i l t i r r nr c j tt rl l a lar rl l li rri er.rnj trigo no fuletric identities sig .i idei]t]t es aoil a a su hr aoid r ae 4 .i clai lci es pptei e i]t ideiit.t alri r r jrlr j,r:1 -l rjn ljnl ian,i l.inr r i rirr irn i ian trn 1-unaunri ;ai angle rr lc l rsr d l forit]ijlas .rla:1 in o

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onlcnns a new honrzorv srxrh eonotl howrno aruron drexel university john wiley sons inc new york chichester brisbane toronto singapore

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mathematics editor barbara holland associate editor sharon smith freelance production manager jeanine furino production and text design hrs electronic text l 4anagement copy editor lilian brady photo editor hilary newman electronic lllustration techsetters lnc typesetting techsetters lnc cover design l 4adelyn lesure cover photo o dann colfey/the lmage bank this book was set in times roman by techsctte inc and prinled and bound by von hoffmann press irc the cover was printed by the phoenix color corp recognizing the impoftance ofpreserving what has been written it is a policy of john wiley sons,inc to have books of enduring value published in the united states printed on acid-free paper and we exert our best efforts to that end the paper on this book was manufactured by a mill whose forest management programs include sustained yield harvesting ofits timberlands sustained yield harvesting pinciples ensure that lhe numben of trees cut each year does not exceed the amount of new growth copyright o 1999 john wiley sons inc all rights reserved no part of this publication may be rcproduced stored in a rclrieval system or fansmilted in any form or by any means electronic mechanical photocopying recording scanning or othenvise except as permitted under s ections 1 07 and 108 of the i 976 united states copydghl act without either the prior written permission of the publisher or authorization through payment of the appropriate per copy fee io the copyright clearance center 222 rosewood drive daflvers ma 01923 508 750-8400 fax 508 750-4470 requests 1 the publisher for permission should be addressed to the pe.missions depatment johnwiley sons,lnc 605 thirdavenue new york ny 10158 0012 212 850-6011 fax 212 850-6008 e mailr permreq@wileycom leriv is a registered trademark of soft warehouse inc mdple is a registered tmdemark ofwaterloo maple software inc marhemaica is a registerd fademark of wolfram research,inc tsbn 0 471 15306 0 printed in the united states ofamerica r0 i i 7 6 5 4 3

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aeour hownno aruror,t h urvo.a anton obtained his b.a trom lehigh university his m.a from the univelsity of illinois and his ph.d from the polytechnic university of brooklyn all in mathematics in the early 1960s he workecl for burroughs corporation and avco corporation at cape canaveral florida where he was involved with missile tracking problcms for the manned space program in 1968 he joined the mathematics department at drexel university where hc taught full time until 1983 since that time he has been an adjunct professor at drexel and has devoted the majority oi his time to textbook writing and acriviries for mathematica associations dr anton was president of the epentl seclion of the mathematical association of america maa served on the board of governors of that organization and guided the creation of thc student chapters of the maa he has published numerous research papers in functional analysis approximation theory and topology as well as pedagogical papers on applications of mathematics he is best known lbr his textbooks in mathematics which are among the most widely used in the world there are currentjy more than ninety versions of his books including translations into spanish arabic portugllese italian indonesian french japanese chinese hebrew and german dr anton has an avid interest in computer technology as it relates to mathematical education rnd publishing he has devcloped pedagogical software fbr teaching calculus and linear algebra as well as various sofiware programs for the publishing industry that automate the production ol fbur color mathenatical text and art for relaxation he enjoys traveling and photography.

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to my wife pat my children brian david and lauren ln memory of my mother shirley stephen girard 1750-1831 benefactor albert herr-esteemed colleague and contributor

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wh n i u gon writing the first edition of this calculus text almost 25 years ago the task though daunting was straightiorward in that the content and organization of a standard calculus course was nearly universal-the chalienge for me at that time was to present the material in a livelier style and with greater cla.rity rhan my predecessors since this calculus text is still among the most widely used in the world i take comfort that the goals i set for myself as a young w ter and mathematician have been achieved howevel times are changing and the era of a standard and unive$al calculus course seems destined lbr the repository of slide rules and three-cent stamps we are witnessing a lot of experimentation with the content organization and goals of calculus-sonre of which has been successful and some of which has not thus my challenge in writing the sixth edition has been to create a text that has all of the strengths of he earlier editions yet incorporates those new ideas that are clearly important and have withstood the objective scrutiny of skilled and thoughtful teachers in preparing for this edition i sought advice fiom outstanding teachers at a wide variety of institutions needless to say i received a diversity of opiniods-some leviewers advised against any major changes arguing that the book was already clearly written and wo|king well in the classroom while others felt that major changes were required to ircorporate technology and rnake the book more contemporary i listened carelully and the lively discussions that followed hclped me formulate my pbilosophy for the new edition many of the specific changes are itemized in the pretace but here are some of the general goals add graphing calculator and cas materials to the text in a way that will allow students who have rhose roois ro use them but that will not prevent tbe text from being used by those students who do not have access to that technology place more emphasis on mathematical modeling and appiications incorporate new examples and exercises that will be neaningf ul to today s students and will more accurately convey the role of caiculus in the real world widen the variety of exercises to focus ntore on conceptual understandilrg through coniecture multistep anaiysis expository writing and what-if anaiysis in addition i wanted to provide some optional innovative materials that would capture the student s interest and plovide the kind of prob!em-solving experience that he or she might find in a research or industrial setting this gave birth to atr exciting set of modules that we have called erp lding tlrc c.tlculus hori o these ruodules appear at the ends of selected chapters and each has an optional internet component that we hope will grcw dynamically over time with input from teachers and students in developing this edition i have stood firm on two principles lhat were adhered to in earlier editions the text material is prcsented at a n lathematical level that is suitable for students who will embark on careers in engineering and science lt remains a primary goal of the text to teach the student clear logical mathematical thinking informal discussions play an imponant notivational role in the exposition and are used extensively but cventually i want the studert to be able to read and understand the language of mathematjcs although tbis edition has many changes and new features they have been implemented in ajleible way that will accommodate a wide vadety of teaching philosophies thus i am confidcnt that professors who have had positive experiences with earlier editions will be comfortable with this revision and i am hopeful that those prcfesson who are looking for a contemporary text with an established history ol success in the classroom will be pleased with the innovations in this new edition sincerely l t r howard anton t

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times the words of a complete stranger are difficult to accept that is why i am about to take this first opponunity to intloduce mysell hopefuily by revealing a bit about myself and how i relate to this textbook may help you find these words more compelling tlt a hello my name is ajay arora and i am an electrical engineering student at mcmaster uniyersity in hamilton canada i too was in your place when i began my entry into the much dreaded field of ca lculus the vast amounts of rate of change and antiderivative problems were overwhelming with a little struggle and hard work i successfully completed that course only to be faced with three more advanced level calculus courses what i am about to write is the unbiased truth on how you can be successful in calculus and how this textbook will assist you on yourjourney i have been a member of the student advisory board for this textbook for over a year now the committee came together as a venture from the authors and publisher to get more student input in the development stages instead of simply focusing on feedback when the book was published after a chapter was completed by the author each student committee member evaluated commented and in some cases recommended altemative approaches these tasks involved lots of special deliveries e-mails faxes telephone calls conference calls and of course a whole lot of calculus but in the end it was a total rewarding experiencehow many times have you asked yourself is math really useful or how about will i ever use calculus in the real world i know i havel this textbook will dehnitely help you answer some of these questions with true applications of the theories you leam the modules entitled expanding the calculus horizon have been included for precisely that purpose every module has been critiqued extensively by the student advisory board and i encourage you to try them not only will these applications of calculus surprise you but they may actually help give you direction in a field that you might want to pursue after cotlege i wish you success in this course as well as the many others you will face during youl college career good luckl sincerely 4pjaiay arora mcmaster univercity board from the student advisory best wishes for success tems ut d tllds dan arndt ufii efiiry of et universr aiav arora mcmasl of chicago fatenah lssa ittyola ijniversin messina uriversin o oklahona lrurie haskell steven e pav allred universiry al i e.i walw state ljniversiry

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aeour thrs eorrroru this i is a nrajol revision in keeping with current trcnds in calculus the goa for this edition is to focus morc od conceptuctl understanding and applicabiliq of the subject matter in designing this edition we worked closely with a talented team of reviewers to ensure that the book is sufficiently leridle that it will continue to meet the needs of those using the last edition and at the same time provide a fresh approach for those instructors who are taking their calculus course in a new direction some of the more significant ohanges are as follows provides extensive materjals for instructors who want to use graphing calculators or computer algebra systems however these materials are implemented in a way that allows the text to be used in courses where technology is used heavily moderately or not at all to provide a sound foundation for the technology material i have added a new section entitled graphing functions on calculators and computers computer algebra systems section 1.3 technology this edition horizon modules selected chapters end with modules called expanding the calculus horizon as the narne implies these modules are intended to take the student a step beyond the traditional calculus text the modules all of which are optional can be assigned either as individual or group projects and can be used by instructors to tailor the calculus course to meet their specific needs and teaching philosophies for example there are modules that touch on iteration and dynamical systems modeling from experimental data by culve fitting applications expository report writing and so forth mathematical modeling mathematical modeling plays a more prominent role in this edition the concept of a mathematical model is introduced in sectiol 1.5 and is used extensively thereafter the horizon module for chapter 5 discusses how to obtain mathematical models liom experimental data in section 10.3 we discuss mathematical modeling with differential equations and in section 1i.10 we discuss mathematical modeling with taylor series the horizon module for chapter 17 develops a mathematical model of a hurricane applieability of calculus one of the goals in this edition is to link calculus more closely to the real world and to the student s own experience this theme starts with the introduction and is carried through in the modules examples ard exercises applications appearing in exercises and exitmples are carefully chosen to be sufficiently sinple that they do not divert time from learning important mathematical fundamentals more extensive applications appear in various horizon moduies earlier differential equations basic ideas about ctifferential equations initial-value problems direction fields and integral curyes are introduced concurently with integm rion and then revisited in more detail in chapter 10 quicker entry to functions chapter 1 begins immediately with lirnctions and the precalculus material that lormed the first chapter in earlier editions has been moved to the appendix for the raader this element is new at vafious points in the exposition the student is assigned a brief task sone tasks are appropriate for all readers while others are x

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preface xiii appropriate only for readcrs who have a graphing calculator or a cas the tasks for all readers are designed to immerse the student more deeply ino thc text by asking them to think about an idea and reach some conclusion the tasks for students using technology are designed to familiarize them with the procedures for using that technology by asking them to read their documentation and perform some tcxt-related computation some instructors may want to make these tasks part of their assignments earlier i-ogarithms and exponentiais logarithrnic and exponential funcrions are introduced in chapter 4 florn the exponent poinl of view and thelr revisited in section 7.9 from the integral point of view this provides a richer variety o1 functions to work with earlier in the text fits in better with the discussions ol modeling and makes for a jess fragmented presentation of the analysis of glaphs however fbr instructors who prefer a later presentation of logarithnric and exponential functions there is a guide for doing this on pages xvi and xvii below option there is a new option lbr introducing parametric curves in section 1.7 of chapter i and rcvisiting the material in chapter 12 where calculusrelated issues are discusscd instructors who prefer the traditional late discussion of parametric equations will have no problem teaching section l7 as part of chapter l2 early parametric or 13 a richer more variety in exercises the exercise sets have been revised extensively to create variety-there are rnany more exercises that include conjecture explomtion multistep analysis and cxpository writing the goal has bcen to put more focus on co,rceptual lutderstarrlirrg there are also rnany new exercises that are intended to be solved using a graphing calculator or a cas these are marked with icons for easy identification analysis of functions the old curve sketching material has been replaced by sections 5.1 5.3 on the aniliysis of functions the name change reflects a mole contemporary approacl r to the nraterial-there is more emphasis on the intelplay betwecn technology and calcultrs and more focus on the problem of findin1 a complerc graph that is a graph that contains alj of the significant features of concem principles of lntegral evaluation the old techniques oflntegration has been renamed principles of integral evaluation to reflcct its more contemporary approach to the material the chapter has beer condensed and there is now more emphasis on general methods and lcrs on tricks tbr evaluating complicated or obscure integmls the section entitled using integral tables and computer algebra systems has been expanded and rewritten extensively supplenrentarv hxercises supplementary exercises chapters have been added at tbe ends of new appendix on solving polynomial equations appendix f enritled solving polynomial equations is new it reviews the factor theorem the remainder theorem and procedurcs lbl finding rational roois many students are weak on this material yet it plays an imponant role in dctcrmining whether a polynomial graph generated on a calculator or computel is conrplete rule of four the rule ol fbr.rr r-efers to the presentatiorl of material from the verbal algebraic visual and nunrerical points of view it is used more extensively in this edition where appropliate.

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xiv preface lnternet an intemet site http www.wiley.com college/anton has been established to complement the text this site contains additional horizon modules and technology materials the site is experimental but we expect it to grow dynamically over time orurr fenrungs flexibility this edition has a built-in flexibility that is designed to serve a broad spectrum of calculus philosophies ranging from baditional to reform graphing technology can be used heavily moderately or not at all and the order of presentation of many sections can be permuted to accommodate specific course needs trigonometry review deficiencies in trigonomeby plague many students included a substantial trigonometry review in appendix e so i have historical notes the biographies and historical notes have been a hallmark of this text from its hfft edition and have been maintained in this edition a11 of the biographical matedals have been distilled from standard sources with the goal of captudng the personalities of the great mafiematicians and bringing them to life for the student graded exercise sets section exercise sets are graded to begin with routine problems and progress gradually toward problems of greater difficulty however in the supplementary exercises i have opted not to grade the exercises by level of difficulty to avoid giving the student a predisposition about the level of effort required rigor the challenge of writing a good calculus book is to strike the dght balance between dgor and clarity my goal is to present precise mathematics to the fullest extent possible for the freshman audience but where clarity and rigor conflict i choose clarity however i believe it to be essential that the student unde$tand the difference between a careful proof and an informal argument so i try to make it clear to the rcader when arguments are informal theory involving 6 arguments appear in separate sections so they can be bypassed if desired mathematical level this book is wdtten at a mathematical level that is suitable for students planning on career in engineering or science computer graphics this edition makes extensive use of modem computer graphics to clarify concepts and to help develop the student s ability to visualize mathematical objects particularly in 3-space for those students who are working with graphing technology there are exercises that are designed to develop the student s ability to generate mathematical graphics student review a student advisory board was actively involved in the development process of this edition to provide information on pedagogical clarity and to advise on the development of examples exercises and modules that students would find interesting and relevant.

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much of the precalculus material has been moved to appendices to allow for an earli er presentation of functions however where appropriate we have included quick summaries of review material in the body of the text the material on logarithmic and exponential functions has been reorganized so it can be covered in the first semester an early transcendental presentatjon therc is a guide on the next page for implementing a late transcendental presentation the first 13 chapters ofthe fifth edition are covered io the first 12 chapters ofthe sixth edition the first 7 chapters of the fifth edition conespond to the first 9 chapters of the sixth edition however the number of sections is about the ssme so there is no increase in the number of lectures required to cover the materinl.the new subdivision rs more natural in that the chapter titles now reflect the chapter content more accurately in the sixth edition as in the fifth edition instructors teaching on the semester system should have no trouble covering material on integration in the first semester chapter 11 on infinite series has been condensed from 12 sections to 10 and the material has been reorganized so that taylor polynomials and taylor series appear earlier this makes it possible to cover these topics without covering the entire chapter the material on analytic geometry and polar coordinates which occupied chapters 12 and 13 in the fifth edition is covered in chapter 12 of the sixth edition lhopital s rule was moved to an earlier position so it can be used to analyze the endbehavior of iogarithmic and exponential functions the two parts to the fundamental theorem of calculus which appearcd in separate sections of the fifth edition row appear together in the same section section 7.6

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