JON ROGAWSKI COLIN ADAMS

THIRD EDITION

ALGEBRA Lines Slopo of the Jioc 1hrough P1 = Cx1, YI J :md P2 = 1"2 nl:

)'2, - .)'I 111= =~-

S lope-intercept equatjon or line \\'ilh s lope JU l'.uld _\-in1etce.J){ b: y =111).'+h

Point-slope equation of l ine through P1 = C.r1. )'I) with slope1t1: y- )'I =111(x - .r 1)

Poinlpoint equa1ion of Line through P1 = (.t 1. Y1) and P2 = (.t1. )'2): )'? - \'I ) ' -)'I =ur(.l: -A' 1) where111= - x2 - .r1

lines of slope 1111 ~ind 1112 are par.1llc:.I if :ind only if 1111 = 1n-z. Line.s of s lope 111 1 and 1111 are peq>endicular if and only if,,, 1 = - ,,:1.

Circles Equation 1)f the c ircle \Vilh c~nler {a. b) and radiu.s r :

(.r _ a)2 + (.\' _ b)i = ,2

Distance and Midpoint Formulas Oistaoce bel\~1een P1 = (.t1. >1) and P2 = (.i:z. Yl):

d = / t:rz - '"l)2 + U'2 - .v1 >2 ,..1.1 1.-p p . (\I + .t2 )'I + )'2) 1~11upo1nl O 1 2

2 .

2

Laws of Exponents :r'11 x11 = :r111+" "'

.Y n1- 11 - = .( ;"

(.ry)n = .t1' y"

.rl/11 = e/i

Cx11')11 = .r'""

Special Factorizations , ?

;- - y - = (X + )')(X - }') xl + yl = (.< + y)(xl - x.v + .1'2) .r3 - .r3 == (x - \')(.:r2 + .r,r + r2)

Binomial Theorem (x + y)2 = x2 + 2ry + y2 (.t - .v>2 = x2 - lx;r + .\2

' 3 ? ' 3 (..t + )'}.> = x + 3x-.' + 3.r.,,- + )' (x - y)'1 = x3 - 3x~.v + 3xyl - y3

_ 1 n(11- I) 2 2 (.v + }'}'1 ; x" + nJtl y + ? x11- )' + . .. + (").:r"-k ,,!~ + . .. + n :rv11- I + v" k . . .

where = --------(") 11(11-J ) .. (11 -k+IJ k I 2 3 k

Quadratic Formula , ' - /,.:I: J1>2 - 4ac LI a., . +bx+c=O. then .t = ., .

-"

Inequalities and Absolute Value ff (I < baud b < c. lhc.n (I < cb. I.ti = x ;f x :;:: o I.ti= - .t ifx _::- Cl

-a 0 II l.rl

Angle Measurement rr r.1d.ians = 180

1T I' = - n1t1 180 180

I rad=-

" s = rl'J (ti in radians)

Right Triangle Definitions . (} opp

-S iil = -hyp

adj cvs(J = -

hyp sin8 opp

tanO=-=-cosO adj

cos(} adj 0016=-=-

sintl opp

sec(}= - 1- = hyp cosO adj

I hyp csc8 -=- --= -

Sill Q opp

Trigonometric Functions

adj

TRIGONOMETRY

"Pl'

Fundamental Identities sin2 0 +cos2 e = 1

, ' I + titll- B =sec'"' 0 I+ cot2b = csc2 ~

(" ) . COS l -(j = SUl0 tan(~-o)=cotO The Law of Sines

The Law of Cosines

sin(-fl) = - sin 8 CO$(-()) = CC>S ()

tan\ - 8) = - tan 8 sin(l> + 211' > = siJl O

cos(O + 2ir) =cos &

c L---:b--\'

sin 0 = ~ r x

cosO = -,.

r cs.co= -)'

r sccO = -

..---i--~ (' = (roo-. 0. r sio 0) a2 = b2 + c2 - 2/Jccos A

tan ll = r x

I. sio. 0 1m -- = I 0~1 (J

x cot8 =- -

)' lim I - t."OSO = 0 u-o 8

(.! 11) (0. 11 (! 11) 1: .. " ., (. If -fI.) ,, ! 90' n - (,-; J?) 1 2 7 2 j 1 1

( -, ') ~,, 120" 61r :! (r' ') ' . ' ' - T I ~ 135" 45"' ~ T 2

l'i ISO" 30"' r. o o

(- 1,0j ,"f 1*1' 300"2:"+ cos.r sin .r -.in(.1; - )') = .t< in .r cm}. - cos.r sin)' oostr+ Y> =

Earl~ Transcendentals JON ROGAWSKI University of California, Los Angeles

COLIN ADAMS Williams College

w. H. FR.HEMAN &COMPANY

A Macmillan Education lmprtn1

THIRD EDITION

TO JULIE - Jon TO ALEXA ANO COL m N - Colin

Publisher: Terri Ward Developmental Editors: Tony Palermino, Katrina Wilhelm Marketil" Manager: Cara LeClair Market Deveiopment Manager: Shannon Howard Exeeulive Media Editor: Laura Judge Associate Editor: Marie Dripchak Editorial Assistant: Victoria Garvey Di re

ABOUT THE AUTHORS

COLIN ADAMS

JON ROGAWSKJ

C olin Ad;uns is the Thomas T. Read professor of Mathematic.' at Williams College, \Vhere he has ktught s ince 1985. Colin rece.ived his undergraduate degree from rvtJT and his PhD from the University of Wi.scort'iin. His re.se~trt'.h is in the area of knot theory and lo\v.dinlensional topc>iogy. He has held various grants to support his research. and \Vti tten numerous rese.;1rch ~1rt-tcles.

Col in i s the authror co-author c>f T/1e K1101 Book. How to Ace Ct1/,:r1f11s: TJ1e .~t"l!el ,...,ise Gttitle, How 10 Ace 1!1e Rest ofCala tltts : TJ1e S1reef\.\1;se G11ide. Riot t tl t ile Ct1/c '

CONTENTS I CALCULUS Early Transcendentals

Chapter 1 PRECALCULUS REVIEW 1 Chapter 5 THE INTEGRAL 259 I. I Real Numbers, Functions, and Graphs l 5.1 Approximating and Computing Area 259 1.2 Li near and Quadratic Functions 12 5.2 The Definite Integral 272 1.3 The Basic Classes of Functions 19 5.3 The Indefinite Integral 281 1.4 Trigonometric Functions 23 5.4 The Fundamental Theorem of Calculus, Part I 288 1.5 Inverse Functions 32 5.5 The Fundamental Theorem of Calculus, Part II 294 1.6 Exponential and Logarithmic Functions 40 5.6 Net Change as the Integral of a Rate of Change 300 I. 7 Technology: Calculators and Computers 48 5.7 Substitution Method 306

Chapter Review Exercises 53 5.8 Further Transcendental Functions 313 5.9 Exponential Growth and Decay 318

Chapter 2 LIMITS 55 Chapter Review Exercises 328

2.1 Limits, Rates of Change, and Tangent Lines 55 Chapter 6 APPLICATIONS OF THE INTEGRAL 333 2.2 Limits: A Numerical and Graphical Approach 63 6.1 Area Behteen Two Curves 333 2.3 Basic Limit LavtS 72 6.2 Setting Up Integrals: Volume, Density, Average Value 341 2.4 Limits and Continuity 75 2.5 Evaluating Limits Algebraically 84 6.3 Volumes of Revolution 351 2.6 Trigonometric Limits 89 6.4 The Method of Cylindrical Shel Is 359 2.7 Limits at Infinity 94 6.5 'M:>rk and Energy 365 2.8 Intermediate Value Theorem 100 Chapter Revie-w Exercise-s 371 2.9 The Formal Definition of a Limit 103 Chapter 7 TECHNIQUES OF INTEGRATION 373 Chapter Review Exercises 110

7. l Integration by Parts 373 Chapter 3 DIFFERENTIATION 113 7.2 Trigonometric Integrals 379

7.3 Trigonometric Substitution 386 3.1 Definition of the Derivative 113 7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic 3.2 The Derivative as a Function 121 Functions 392 3.3 Product and Quotient Rules 135 7.5 The Method of Partial Fractions 398 3.4 Rates of Change 142 7.6 Strategies for Integration 407 3.5 Higher Derivatives 151 7.7 Improper Integrals 414 3.6 Trigonometric Functions 156 7.8 Probability and Integration 425 3.7 The Chain Rule 159 7.9 Numerical Integration 431 3.8 Implicit Differentiation 167 Chapter Review Exercises 440 3.9 Derivative.s of General Exponential and Logarithmic

Functions 175 Chapter 8 FURTHER APPLICATIONS OF THE 3.10 Related Rates 182 INTEGRAL AND TAYLOR

Chapter Revie\v Exerci.ses 189 POLYNOMIALS 443

Chapter 4 APPLICATIONS OF THE DERIVATIVE 193 8.1 Arc Length and Surface Area 443 8.2 Fluid Pressure and Force 450 4.1 Li near Approximation and Applications 193 8.3 Center of Mass 456 4.2 Extreme Values 200 8.4 Taylor Polynomials 465 4.3 The Mean Value Theorem and Monotonicity 210 Chapter Reviev1 Exercises 476 4.4 The Shape of a Graph 217 4.5 t:H6pital's Rule 224 Chapter 9 INTRODUCTION TO DIFFERENTIAL 4.6 Graph Sketching and Asymptotes 231 EQUATIONS 479 4.7 Applied Optimization 239 4.8 Newton's Method 251 9. l Salving Differential Equations 479

Chapter Revie\v Exercises 256 9.2 Models Involving y' = k (y - b ) 487

PREFACE

ABOUT CALCULUS On Teaching Mathematics J consider m )''self very lucky l have a career a ..... a tc--:.-tcller and practitione.r of mathemalics. Wilen I \va..; young. I de.cided I wan1ed l Lx. a \\.Titer. J lved te-lling stories. Bul J \Vas also god at 1nath, ancl nee in college. it didn. t Lake me lc>ng to beco1necnan1o red \Vilh it. I lved the facl Lllal success in 1nallle1natics does llC>t depctlCI n your pre;entatic>n skillr of n1athen1atics ... t\00 J sn realized Lhat Leaching 11i:.1the n1atic.'i. i.'i. a.bc>ul Lelling a s1ory. The goal i.'i. t explain to studenl'i in an inLriguing manne.r. al Lhe 1ighl pace, and in as clear a \vay as possible, ho\v n1all1c1nalics \Vrks and \vhal iL can de> for )'Ou. I find 1n..1the1nalics ilnmenscly be.a ULiful. J \VanL student.'i. t fee.I ll1al \vay, too.

On Writing a Calculus Text J had alv.:ays Lhoughl 1 n1ight wrilea ca.lculn'i. text But Ll1nt i..~ a daunting task. Tliesedays, calculus boks ave.rage over a Ll1oll'i.and pages. And I would neOO lo convince myself that J 1-.ad smethlng to ff er ll-.al was different enough from \vl1al al~-tdy appek for \vhich J alrealy had g re.al respect Jon's visic>n of what a calculus bcx>k should be fi l very closely \Vilh n1y \VO. Jc>n believed Ll1at \V \Ve say it i.~ t1s import.1nL a.'\ wll:.1t \Ve say . . A.hhough he insisted on rigc>r al ;.111 titnes. he also \vanted a book lhal \Va.~ \Vti tten in plain English. ;.1 book Lhat could be re-ad and Lhat \voukl entic.e s1uclcnl'\ to ~-td furll1er and le.:.1rn n1c>rc. tvlc>rever. Jc>n strived t cre.:.llea text in \Vhich expc>sitic>n. graphics. and layoul \VC>uld \Vrk togclher to enhance ;.111 facets of a studcnts calculus experience ..

In \vriling his book, Jc>n paid spe.cia1 altenlion t cerLain a.'\pccl'i of lhe tc.xt

1. Clear, accessi'bleexposition that antic-ipate."i and addresses s tudent d ifficultie,.'\. 2-. Layout and figure~~ tll:.tt communicate the fto\v f idea.'i. 3. Highlig hted feature.s that empl1asizeconcepts a11d 1n.athe1natical rea.'ining: Conceplual Insight. Gr..tphic.al Insight, . .l\.ssumptioll.'\ Matter. Re1ninder, and Hi..~ trce conceptual understanding, and n1o tivatecal culll'\ lhrugh interesting ap1)lications. Each section also contains exercise,~ that develop additinal insight.~ and challenge student~ to further develop their skills.

Coming into Lhe project f cre.ating the third edition, I \Vas sme\vhat apprehensive. Here \Va. an already excellent book that had ilttained the goals .se.t for it by iL'i autl10r: First and fore1nost, I \Vilnted Lo be sure that I did it no harm. On the o ther hand, J have been te.:.1c.hing calculus no\v for 30 years, and in that time, I have c.01ne t some cnclusions about \Vhat de.'i and does nol \vork \veil for students.

As a mathe1natician, J \Vant to make sure that the theore1n s. pn>ofs. argumenls and deveJop1nent are cotToct. There is n place in mathematics for sloppiness of any klnd. As a t~1cher, J \Vant the matetial t be accessible. The book should not be \Vtitten at the n1athe.matical leve.I f the instructor. Stude.nts shc>uld be able to use the book to learn the n1a1c.rial, \Vi th the help of their itl.'ilJuctor. Working from the high stiuldard that Jon set. J have uied hard to 1naintain the level of quality f the previous edition \vhile making the c hanges that I be.lieve will bring the boo k to the next level.

PREFACE vi

Placement of Taylor Polynomials Taylor polynomi,~s appear in Chapte r 8, before infinite series in Chapte r 10 . The goal here is l pre.."ient Taylor pofynon1ial.s as a natural extension of lit-.e.ar approximatic>n. When teaching infinite se.ri:e..-;, the pri111ary focll'i is on conve.rgence, a topic that many st ude.nL'i find c hallenging. By the ti1ne \Ve llf po\ver se1ies, students are ready l tackle the issue.~ involved in representing a funclion by its Taylor series. They can then rely n their previous work wit.11 Taylor polynon1ials and Lhe e1Tor bound frotn Ch..1pter 8. However, Lile section on Taylor ~)()lyno111 i.a.ls is \vritten sc> I.hat )'OU can cover thi.'i topic togetl1er \vitl1 the mate1i~1ls on infinile serie..-. if this rder is preferred.

Careful, Precise Development W. H. Frre1nan is com1n fLted to high quality and precise textbooks and supplernent..'i. Fron1 Lllis project's ir.ception and Ll1roughout its devek>pmenL and _productin. quality and precision have been given significant prio rity. We have (n place unparallek!d _procedure.'i to en.sure Lite accuracy of Lhe text

Exe.rcise.'i atld E.xampk!.s Exposition Figures Editing Cmpositlon

Togetller, these procedwe..'i far exceed prior indu.sby st~uldards to safeguard the quality and precision of a c-alculu.s textbook.

New to the Third Edition There are a variety of change..s lhal have been iLnplemented in Lltis edition. Fllo\ving are so1neof lhe most impo11anL

MORE FOCUS ON CONCEPTS The emphasis has been shifted to focus le.o;., on Lhe memo 1iz.ation of specific formulas, and mreon underst:..1nding tl-.e unde.1iying encepls. Memo ... 1iz.ation c;.111 never be completely avoide~ but it Ls in no \Vay the c1ux of calculus. StuclcnL~ \Viii remembc.r h'v to apply a procedure or technique if Lhey see the lc>glc.al progre ... s..sin Lhal generates it /\1ld lhey Ll1en understru1d Ll1e underlying concepts ralller lhan seeing tl1e topic asa black box in \Vhich you inse1t nu1nber.s. Specificexa1nples include:

(Seetion 1.2) Removed the gene ral fonnula for the complerion o f a square and inste.ad, emphasized the method so s tudcnt..s need not 1nemrize the fo11nula. (Sectk>n 7.2) Cha.nged the rnelhods for evaluating trigonometric integr~tls tc> fcu.s n technique .. s lo apply rather than fo1mula.s t me.morize. (Clutpler 9) Discouraged the me morization of slutio ns of specific types of d iffer ential equations and in.'ite-n. (Sectk>n 12.2) Dec-reased number of frmulas for parametrizing a lire f mm t \V tc> ne, as tl1e second C..'ln easily be derived fmm the first. (Sectk>n 12.6) De-e.mphasize.d the men101i z.alin of the various fc>rmula.s for quadric surfaces. Instead, moved the fc>eus to slicing \Vith plane.."'i t find curve.s and u.sing tl1ose to dete1mine the shape of the surface. These methods \viii be u.seful regardle.s.s of the type of surface it is. (Sectk>n 14.4) Decr~-tsed the numbe.rf essential fo1mulas fr line.ar apprxi1n.;.1tion of functlons of l\VO vati ables f min four t t\VO, provid ing the background Lo derive the o thers from Lliese.

CHANGES IN NOTATION There are numerous nota rional c hange.,. Some were made to bring Lhe notation 1nre into line \vith s u1ndard usage in mathe1natics and olhe.r fields in \vhich 1nalhe1natics i.s applied. Sonle \Vere implemented t 1nake il ca..sier for sludenl'i to ren1ember tllf' 1neaning f the 110tatic>n. So1ne were 1nade lo help m~tke thecorte.sponding co1x:epts Ll1at are represented mre ttansparenL Specific examples include:

'till PREFACE

(Section 4.6) Presented a ne w notation for graphing that g ives the s ign s or the fi rs t and second derivative and Lhen sirnpfe sytnbc>ls (slanted up and 00\vn arr\VS and up and d o\vn u's) to help Lhe s tudent keep track of when the g raph is i 11c.r~t'iing or decreasing ~uld concave up or concave do\vn over the given interval. (Section 7.1) Simplified the notation for integration by part' and provided a visual 1nethod for re1ne1nbe.rlng it. (Chapter I 0) Chan goo names or the variou.' tests for conve rgence/divergence or infinite serie.'i t evc>ke the usage of the te.st and thereby 1nake it c.a.'iicr for students to re1ne1nbe.r the1n. (Chapters 13- 17) Rather than u.' ing c(r) for a path, we consiste ntly switched Lo the vector. valued functin r ( I). This al.s alk>\ved ll'i to replace tis \vith d r a .. 'i a d ifferential, \vhich means there is les .. 'i likely l be confusion \Vilh ds,dS and dS.

MORE EXPLANATIONS OF DERIVATIONS Occasionally, in the previous edition, a re->ult \Vas given and verified, without motivating where tile derivation c~une f rorn. I believe il ls i1npo11anl for students to understand llO\V sorneor.e 1night come up with a particular result, thereby helping them Lo picture how they might themselves one day be able Lo derive resulLo;;.

(S ection 14.4) Developed theequation or the tangent pkrne in a manner that ma kes get>1net1ic se.nse. (S ection 14.5) Included a proof of the fact the gradient o f a function f of three variables is 011hogonal lo the surface..s that are the level sets off. (S ection 14.8) Gave an intuitive e xpkrnation for why the Method of Lagrange fVlultipliers \VOrks . (S ection 15.5) Developed the cente r or mas.' formula' by first dtscussing the one d imensinal case of a see.'\.a\V.

REORDERING ANO ADDING TOPICS T he re weresme specific ,,,a,,.angement' among the sections ai1d addiLio1t.s. These ioclucle:

A subsection on piece\vise-defi ned functions has been added t Section 1.3. T l., section on implic it d iffe rentiation in Chapter 3 (previously Section 3. IO) has

been rnved up to become Section 3. 8 ;_u1d has absrbed the. prevtous Section 3.8 (in verse functirts) so that itnplicit differentiation can be applied to detive the. various derivatives as necess.:1ry.

Tl-.e .section n indefinite integrals (previusly Scctin 4.9) has been 1n oved fn>tn Chapte r 4 (Application.s of tl>e Deiivative) Lo Chapter 5 (The lntegral). This is a rnre naturt1l place1nent for it.

l\ ne\v section on choc>sing front amongst the various methods of integration ha .. s been added Lo Chapter 7 .

l\ subsection c>n choosing the appropriate convergence/dlvergetlCe te..o;;t has been added to Section 10 .5.

.l\.n explanation of hc>\V lo fi 1xl inde.finite limits using p\Ver serie.s ha .. s been added Lo Section 10 .6. The-de.fl nitions of d ivergence and curl have been mc>ved front Chapter 17 lo Scctin 16. 1. This allo\VS us to utilize them at an appropriate earlier point in the text. A lis t all of the different type_, of integral' that have been introduced in Chapter 16 ha' been added Lo Sectio n 16.5 . . A. subsection on the Vector Form of Green's Theorem ha .. been added lo Scctin 17. I.

NEW EXAMPLES, FIGURES, ANO EXERCISES Numerous examples and '1ccompanying figures have been added l> cla1ify concepLs .. A. variety f exercises have also been added throughout the Lext, ~Ja.rticul~trly \Vhere ne\v applic;_ttions are available o r f wther conceptual dcvek>ptnent is adv~1ntageollli. Figures marked \Vitl1 a GI icc>n have been made dy1l:ltnic and can be accessed via lau11c/1Patl . . I\ se~ctin f the.lie figures also includes brief llltorial videos explaining the concept'> al \YX>rk.

ONLINE HOMEWORK OPTIONS

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MuhivariablelSBN: 1464 1-7 189-0 Test Bank Compu1e1ized (CD-ROM), ISBN: 1-3 190.09395

Contai11s \VC>rked--oul solutions to all odd-nu1nbered exercises. in the text

Includes a comprelie11sl ve .set of muhiple-choice tc..'it ite.ms. Software Manuals

Instructor's Resource Manual Prvide~'i s;.unple course oullinc.s, sugge .. sled cla.s..'i Lime, key points, lecture mate tial. discussion tc>pics, class activities, \Vork sbeets, project-;, a1xl questions t accom1Jany the. Dy11amic Fig

ure~'i.

Maplen ' and Mathen1atica soft,vare 1n;.u1ual'i serve a.'i basic introduclions to popular mathcn1atical soft \Vare pt ions.

ONLINE HOMEWORK OPTIONS

W LeamlngCurve

GoiutionMaster

Ow ne\v curse space, Lau1lChPad, co1n binc,s an interacLive eBok \Vilh higllquality multimedia contenL and ready .. made asse.ssment options. i1x:luding Lec1rningCurve adap Live quiz.zing. Pre.--builL, curated uni Ls are e.a.sy to assign or adapt \Villl yur O\Vn mate1ial, such as retn each o ll-.er. U.se LaunchPad on its O\Vll r integrate it \Vi th your sch0l's learning 1nanagemenl system so your clas .. -s is al\\"ays on the san1e page. Contac.l your rep Lo 1nake sure yU have access.

1\ssets integrated inLo LauncltPad include:

Interactive eB(M>k: Every Laur.chPad e Book cn1e.s \vith pc>\ve1ful study tols for stu dents, video and multimedia cntent, a1ld e..1Sy custn1i'z..alion for instructc)rs. Students can se quizze....;., activities, and other resurce..--s.

LearniogCur\'e p1uvides students and inslrucleu'S \vith po\verful adaptive quizzing, a game.-like fonnat, direct links to l11e e-Bcx>k> and instant feedback. The quizzing syste1n fe..a lu1-e...; questions l students' re~sponse.s. pro viding maLe1ial al different difficulty levels and topics based on sLudenL lJe1fo11nrs to nmnipulate \ving insuuctors to generate a solution ti le for any set of ho1nework exercises.

1 FEATU RES

FEATURES

Conceptual Insights e1lCourage students to de'll-elop a cnceptual unders tanding f calculus by explaining impo11ant ideas clearly but infc>r1naHy.

WebAssign. P'~ntwn

'"'r,,, \l'l'has..\ ign.1H.'t/l'reeman.c:m Web.,lulions. WebAssign Pre.miu1n also offers acce.s .. i; to re.sources, including Dy1-.:_1mic-Figures, CalcClips \Vhiteboard tutrkils, and a "Sho\v My Work" fe..tture. Jn additK>n~ Web.,pe.rties and the underlying concepL~.

( A) U.1gc so:ond ck::tivulivc: T:lngcn1 lines turnropidly.

1EJ RGURE 3

(8) Snwl~t S(X'ond dcri\.-ativc: T:ingcn1 lines tum slo.,,.,ly.

(C) So.--ond Mi...-a th~ is zcn>: Tangcn1 line doc!\ 001 ch:ingc.

Ch. 3. p. 1S3

Reminders ;.ue 1n.:1rgin ootes Lllal link LJ1ecurrenl discussion to i1npo1tant concepts inLrdoced ear lier in the text to give students a quick revie,1.: and make cnnections \Vilh related ideas.

" lttMt#Jl1tU'J.ffuf lrk11tiriesl l ~inr - 2{1 -co~lr>

I te"T i-(J + Ctl:clr>

:J1112x - 2.1mxai.1.Y

CUI lx .:.o? ,Y - .dn: .Y

FEATURES .d

EXAMPLE 3 0n \\~could :.pply the l\.'({11a100 fut11rul : 1n::ttgin, .,..;: can :'Iii incgmcc~ .l .OOc.sin

ITTglhc fo11o .... 1ni:

j ' .t sin2.r .c I J111 :rd.t =? - - + C =? - ;31n.rC

Solution LC! /C.~) =.tl -8Md gl:l') = .l.~ + 2x - 20. BO!h / Mdgsredilfctenti tlblc tulil j (.t)/g(.t ) " intcrniinmc or 1ypcO/O ni er = 2 b:'.cau:i:'. j i2) = g(2) = 0:

Nu.mett110f: j(2) = 23 - I = () Denn'li.n:ttoc g(l ) = 2J + 2(2) -10 = 0 Cb riat eti tff!'r.l!tiak lhr. qwlir.nt flt111:toa1J

'f f(.v )/gtv). Funhi.':miotc. g1Ll) =tr'+ 2 I:. no1t1.ero new .t = 2. so L' i'f6phal's Ru applies.. \Ve mtiy rephi.:c the ntu1~m1or tllld ~min:llot by thcit d\!l'i \'ati,cs 1oobu:tin

Ch.4, p. Z24

Historical Perspectives are brief vignette~~ lhal place key discoveries a1xl conceptual advtulCes in Ll1cir hisliel. W" gel

00

s- r + 1r + 1 r + - r"' 1 - 4 r .- 4 i6" - t....,F-j

-0 l'n.e 1nod..-m l41d y b ::ad Jm~a t.e

ti ACKNOWLEOGMENTS

Assumptions Malter use.~ short expla1ln.'i indicating problems I.hat require ll-.e studenl to give. a \Vti lten

response~ r require ll-.e use of technology Ill Chapter Review Exercises offer a cx>mprehensiveset f cxe.rcises closely cordinaled \vill1 t.l1e chapter 1naterial to provide additional problems for self Study or assigntnenls.

ACKNOWLEDGMENTS Colin Adams and W. H. Freellk-1n and Con1pany :ne g rateful to the 1nany i11sLructors fmn1 across the lJniled Slale.'i and Canada \Vho have offered con11nenl'i tl1Jote Cinn111u11ity Ci>llege; David Oernpsty . .lack.r1J11~i//e .'>tate Uni~er:fity ; Edwin Sn1i th . .lac:k.fm1vil/e .Statt! UniverJit.r: Jeff Dolate Polytech11i1. Uni~er.Uty. P1J11K111il: Chari~ Utn1. CalijOntill Slate Uni~ en it)~ /Jaker.rfield: Da \'id t\1cKay, Ci1/if 1Jrnia State lJni~ersit): LJ111J( Beac:h: Melvin Lax, C11/ffon1ia State U11i~crsity. /JJug Beach: \Val-late-A. Et 1e.rheek, Calf/(Jruia .Stilte Uni~er.lity, .Sac:ru111en1

Pe.ter -Stillhis . Glendale C'1n1uu111i1y Ci1/Jege: Douglas D. LIO)d. Ga/den lVe.rl C'11lege: Thc'Jluas ScaR'.lina. Golden \\~.rJ ClJIJege: Kri.. lin HarttOrd, LilllJJ Beai./J Ci1y Ce: EdutlfdO Ari.s1nendi4 Pardi. Orange C'1a.r1 Col lege: r..-1i1chell Al\'e.e; Eleanor Lang Ke-ndrick . San .lo.te City College: El i.t.abeth Hoclto1J 'if the ft1ines : J in1 ThU111a.-i. C

"' ACKNOWLEOGMENTS

Jone..:;, J\liddle Ge

The Univer:rily

n; ACKNOWLEOGMENTS

Tech Uuiver:rity : I. Wayne- l..e\vis, Texa.f Te

Fuoctio1'6 lhn1yield1hc nmoun1 o f seismic nctivi1y !lSnrepe..ating. Fr exan1ple,

3 8 = 0.375. 1 'i = 0. 142857 J 42857 . .. = o. 142857 l( = 3. 14 1592653589793 .. .

The nu1nber ~ is represented by a finiLe deci1nal, wherea..'i ~ is represc.nled by a repea1;ngor period;c decirnal. T l--.e bar over 142857 incliI for the phr;~o;e "belongs to." Thll,,

a R reads "a belongs to R"

The sel of inlegers is cmn1nly denoted by Lhe letter Z (this choice co1nes from t11e German word illhl, meaning '\>Umbe r"). Thll,, Z = j . . .. -2. - 1. 0. 1, 2, ... }. A wtmle number is a nonnegative integer- that i.'i. one of Lile numbers 0, I. 2 . ....

1\ re.al number is called ratitn the follo\\ingexce.plion: Every fini tedeci1nal is equal lo an infinite decimal in \Vhich ~1e digit 9 repe

2 C HAP TE R I PRECALCULUS REVIEW

- 2 - I a 0 2 b FIGURE 3 T he.dii;muce frtlUle open interval (.r : 2 < .r < 3) or it could 1nean 111e point in the .r)-plane V/ith .r = 2 aad .r = 3. In general, l11e n'Jeaning will t:e apparent frotn 111econteXI.

AGURE 5 Oosed half-iofinite irltetval s.

_, () AGUR 6 T he interval (- r.r) = {x : lxl < r) .

'

c-' c

'

'

r+' FIGURE 7 (a. b ) = (< - r. int tlnd i..s pen olhe.l'\vise (Figure 5):

[a,oo)= {x :a :;x ), (-oo,b] = {x : x < b)

" b

((1. 00) 1-oc.bj

Open a1-xlclosed inter\.als 1nay bedesc-ribed by inequalities. Forexatnple, the interval (-r.r) is described by Lhe inequality lx l < 1 (Figwe 6):

I [t i

J J

I() lJ FIGURE 8 Tile interval 1_7. 13] i.o; desciibed by l.r - 101 :!; J.

Jn Example 2 we use the notation u to denote .. union": The union A U 8 of sets A and B consists of all elemetlts tf)at belong 10 either ;\ 0t 8 (or to both).

0 I l)o-ooo - 2 0 14

FIGURE 9 The set S =Ix: ltx - JI> 41.

Tl1e tern1 "Cartesian" refers to tJ)e Frooch philosopher and mathematician Rene Dascartes (1596-1650), wl>ose Latin nan'!f! was Cartesius. He is credited {a!011g with P1e1re de f(!(lnaD with 111e invm1tion of anatyuc geo1netty. Jn 111s great work La G~om~ttie , Descartes used 1l1e letters . r . ) .. ~ for u11kncwns and a. b, t: for constants, a conve11tion thaI has been fol/CN;gd ever since.

FIGURE 10 Rectangular coordill)'$ICO\.

.)

, .

-1---- 4} in 1e1m sof intervals. Sn J l L~ easier Lo consider the op1x>site inequality I ! .r - 3 I S 4 fi rsl. By (2),

I -x -3 54 ~ 2

J -4 < -x -3 < 4

- 2 -

J - I < - x

4 C HAP TE R I PRECALCULUS REVIEW

.I'

'

b lo, b)

a

FIGURE 12 Ci rcle .,..,i1.h equa1ion ,, - )2 + (y - b)1 = r1.

(x. ) j

A function f : D - Y is afso called a "1nap." Tl1e sets D al)d )' cao be arbitrary. For ex.a1nple, vte can define a 1nap fro1n ihe ser of litAng people to the set of wJ)o/e numbetS by 1nappl11g each petSoo to f1is or het year of birth. The range of this map is rl)e sel of ~&'Sin wl1ic/1 a living perSon v1as born. Jn 1nultivariable calculus. the d

- -Y'---11< I. I) I I

I ----.-J:---l.h f till\ 1.he Vel'lical Line Test., so ii is not Lhe graph of a ruriction.

SECT I ON I. I Real Hu1nbe1s, funttions, aBd Graphs 5

EXAMPLE 3 Find the roots and ske tc h the grnphof.f(x) = x 3 -2.r. &)lutk>n Firs l, \Ve.slve

3 ' x - 2x = x(x- -2) = 0

T he roots o f.f are x = ()and x = +J2. To s ke tc.h the graph, we plot the roots and a few values listed in Table ) and join them by a c urve (Figure 16).

TABLE 1

- 2 - 1

0

2

- 4 I 0

- 1 4

-4

FIGURE 16 Go~ph of f(x) = .rl - 2.r. Functions arising in applications are noL alv.:a)''S g iven by frmula.s. Fr exatnple,

data collected fmtn bse.rvatin r e xpetime nt define.functions for \Vhlch there may be no exact formula. Suc h functions can be d isplayed e ither graphically or by a table o f values. Fig ure J7 and Table 2 d isplay data collected by biologist Julian Huxley ( 1887- 1975) in a study f the antler \veight \V f male red dee.r as a futlCLin f age t. We \vill see that many f the tools fro1n calculll'i can be applied to functions crt'itructed from dala in lhis \Vay.

8 7 6 5 4 3 2

()

AntlcrlA"Cighl H' ( kg)

/ '\

-

/ .

f-- /

.

I I () 2 4 6 8 10 12

Age I {yc~U'S) FIGURE 17 rvtate red deer shed 1.hei r antlers e\fery 'vinter and regrov. 1.hen1 in 1.he spring. T hisg.1aph ~hO\' .. saverage anT ler \Veig.h1 a.'> a f1111cti 011 of age.

TABLE 2

I ()'eao

& C HAP TE R I PR ECALCU LUS REVIE W

) '

+l------x

(A) I ncreasing

FIGURE 19

!

We Sa)'' thal f is mn: Graph iss y1n 1netric about the )~ .. axis. This 1nere~ }1 is neither even nor dd.

EXAMPLE 5 Using Symmetry Ske tch the graph o f f(x) = , I 1.

x - + S olution The function f L' positive [f (x ) > 0.1 and e ve.n [f ( -x) = f (.r ) J. There.fore , the g raph lies above the .\ .. axis and is symntetric \\tith respect to the )'axis. Fwthennore~

Remembe< that f(x) + c and f(x + t. The graph of.)' = /(:r) + t is a verrical traos/ation and.'' = f(:r + ':) a horizontal translation of t11e graph of )' = f(x).

S ECT I O N I.I Real Hu1nbe1s, funttions, aBd Graphs 1

f is decreasing for .r > () (bee.au.~ a larger value of .r 111. 0 and do\vn\vard if c < 0. Horizontal lran.lation y = .f(x + c): Shifts the g raph by !cl units horiwm ally, to the right if t < 0 and t uniL'i t the left i f t > 0.

Figwe 22 shows the effect of translating the graph of .f (x) = J /(x2 + I) veit ically and horizont..1lly.

)'

2

- 2 - 1

I (A) )'=f(.l) = ..,-.t -t I

FIGURE 22

2

; Shifl I unit

upward ?

I -+--+---+---+--+-. - 2 - 1 2

I ( B) \'=}fxJ+ I =-+ I _.2 + I

Shifl I uni1 2: to 1he left

-J - 2 - 1

(C)1=}\x+ I) = 1, '+ I

EXAMPLE 6 Fig ure23(A) is the graph of .f(x ) = x2 , and Fig ure23(B) is a hmizomal and ve1t ica l shift of (A). What L' the equation o f graph (B)?

)'

4

l 3

2

--+--+-~..:......._ ____ x

- 2: - 1 2 l

(B) lil) f lGURE 23

Solulkm Graph (B) is obtained by shifting graph (A) I unit ID the rig ht and I unit down. We can see this by observing that the point (0. 0) on theg1'1ph of .f ''shifted to ( I. - J). Therefore, (8) '' the g raph o f g (x) = (x - 1)2 - I.

8 C HAP T ER I PR ECALCULUS REVIEW

)

2

- 2

- 1 .Y= - 2j(.t)

FIGURE 24 Ne-gaLi\'C ve1'Lical .scale f I, the graph isexpanded ve1tically by the factor k . If 0 < k < I, the graph is compressed ve rtically. When the =~e faclor k is nega tive (k < 0 ), the graph is also reftec1ed acros.s Lhex axis (Figure 24).

H orizontal scaling y = f (kx ): Irk > I. the grnph is compressed in the horizontal d irection. If 0 < k < I. lhe g raph is expanded. If k < 0. then Lhe graph is also reflected acrss the. )~axis.

The amplitude of a function is half Lhe difference bet\veen iL\ gre-4:tte.sl value and its le:.1.st value, if il has both .a gre.atest v~1lue ~tnd Jea.\ l value. Thll'i. ve11lcal .scaling change.\ Lhe ampliLUde by the factor lk (. EXAMPLE 7 Sketch ti-,, graphs o f f (x ) = sin(rrx) and i1sdilates /(3x} and 3 f (x ). Slutin The graph off (.t) = sin(rr .t) is a sine curve \Vilh peri

S ECT I ON I.I Real Hu1nbe1s, funttions, aBd Graphs 9

Distanced between (x1. yi) and (.r2. )~):

d = /2 + (y2 - )'1 >2 Equation of c ircle of radius r \Vi th cente r (a. /J) :

(x - a)1 + (y - b) l = ,.2

A zero r roo1 fa functin f is a number c such that f(c) = 0. Vert ical Li1-.e Test: A curve in the plane is the graph of a functic>n if a1lCI only if each vertical line.r =a intersect.'i the curve in at most ne point.

Jncre:.-tsing: /(.q ) < f(x1) if.r 1 < .\"2 Nndecreasing: f(.r 1) < f(x1) if X i < .'(~ Decreasing: /(x1) > f(x1) if .r1 < .r2

Nonincre~t'iing: /(.q ) 2:: f(x1) if .r1 < .\"2 Even function: f (-x ) = f (x) (graph is symme tric about the )~axis). Odd function: f(-x) = -f(x) (graph is symme tric about the origin). Four ways to transform the graph of/:

/ (.r ) + c Shifts graph vertically Jc l units (upward if c > 0, downward if c < 0) f (x + c) Shifts g raph horizontally Jcl units (to the rig ht if c < 0. to the left if c > 0 ) kf l t ) Sc.ale~'i graph vertically by faclr k~

ifk < 0, g raph is reftecled across.r -axis f (kx ) &ales graph horizontally by factor k (compre-o;.o;es if k > I);

if k < 0. graph i.~ reflected acro.s.s .v..axi.~

1.1 EXERCISES

Preliminary Questions I. Gi,eaoe~an\pleof nurnbers a aod b such that a < band !al > !bl .

2. \Vhic.h nun1bers sa1i.~f)1 lal = a? \\'llich sati sfy It.ii = - a? \VhaL about I- al = a?

3. Give an e.' lal + lb!'!

5. \Vhat are 1.he cootdi nates of the point I )'ing at the i 1lteti:OCl ion of the lines.r = 9 and >' = - 4?

Exercises I. Use a calclllator 10 find a rational 1lulllber r such that

lr - 1'21 < 10- . 2. \Vhich of (aHf) are 111.Je ford = - 3 and b = 2?

(a) a < b (b) lal < lhl (c) ab > 0 I I (dl 3a < 3b (el - 4a < - 4b (0 ;; < b

In Exen:i.1e:r 3-8. ex11reJ:.i the inter\'ill in ter111.1,~f an inequali I)' in\'tJ/\i11g absolute tJJJ1e.

3.J- 2,2] 6. 1- 4. cYI

4. (- 4, 4)

7. 11. 5J

5. (0, 4)

8. (- 2, 8)

6. In \\'ltich

10 C HAP TER l PRECALCULUS REVIEW

/11 Exen.'ises 19-22. de.rtribe the :wl aJ a u11io11 of finite or injinite in-ter~al:t.

19. {,, : l.r - 41 > 2) ? 21. {x : l.r - - 11> 2)

23. rvta1c.h (aHf) \vi1h (iHvi).

(a) a > 3

(c) la - ~I < 5 (el la - 41 < 3

(i) a lies 10 1he rig.ht of 3. (ii) a lies bet" een I .and 7.

20. !x : 12.r + 41 > 3)

n . {.r : 1.r2 + 2.rl > 2)

I (b) la - 51 < -3 (d ) lal > 5

(01 ~ SI 2. (v) a is less than 5 uni t"'> froo\ } .

(vi) a lies either 10 1.he lefl: of - 5 or to I.he right of 5.

24. Desc1i be lr : -"- < oJ a..; an in1erval. f/in1: Consider 1.hesign .r + I

of .r aud .r + I indi,1id ually.

25. Desc1i be fx : .r2 + 2.r < 3) a.-. an i ntervaL flint : Plo1 J' = .r2 + ? ' -" -.),

26. Desc1i be 1 he set or real 1n.1nlber.s SlltiSf)ing l.r - 31 = l.r - 21 + I a.c; a hal f-infinite interval.

27. ShO\\' 1.ha1 if a > b . .and a, b -:/; 0, 1he11 b-1 -> a - 1, rxovided IJlat a a.ndb have the sa11lesign. \Vhat happens if a > C) and b in1s and C

61. De1e1'l'll ine \\'he1.hei tlle function is evetl , odd, or neither. I I (a) [(1) - (b) g(1) = 2 1 - 2- 1

- 1 + 1 + 1 1' - 1+ 1 (c) G(O) = sinu + oosO (d) H(O) = sin(01)

62. \Vrile /(:r) = 2x4 - 5.r-3 + 12 .,.2 - 3.r + 4 a.~ the s u.01 of ail C\'Cll and all odd functi on.

( I -x) 63. S ho\v tha1 f(.r) = In 1 + .r is an odd function. 64. S1ate \Vhetller 1he function is increa.t.:i ng, decrea1'i ng. OJ' nei1 her. (a) Su1face area of a sphere as a ruoction or iL1; radiu.s (b) Teo1pera1ure at a 1X>int on 1t~ equator a.t.: a functit)ll of till)e (c) Price of an airline ticket as a function of 1.he price of oil (d) Pre.f\Sure of 1 he ga.t.: in a piS101\ as a f unctio1l of \'OI unle In E.ren:ise.f 65-70. let f be 1he,funt:ti1111 :'111111'11 in Figull! 27. 65. Find tlh~ do1nain -and range of/.

66. Sketch ~iesraphsof ,r = f(.r + 2)and y = f(x) + 2.

67. Skeich the zraplt< of y = /(2.r). y = f (t x). and y = 2 f(x ). 68. Sketch ~>e graphs of y = f( - x) and y = - f( - x).

69. Extend 1.he graph of f Lo [- 4 . 4J so tJta1 ii i..'> an even function.

70. EXIt-nd 1.he graph of f 10 [- 4 . 4 ] so Lhat ii i..~ an odd function.

)

I 2 3 4 RGUR 27

71. Suppose tJlal f ha.~ doo\ain 14, 81and range [2. 6J. Pind tJle do1n.ain and range of: (a) y = f(x) + 3 (b) y = f(x + J) (c) ,r = f(Jx) (d) ,r = 3/(x)

72. LCJ f(x) = .t 1. Sketch ~le graph over [- 2, 2] of: ( a) y = f(x + I) (b) y = f(x) + I (C) )' = f(5x) (dj y = 5 f(.r)

73. Suppo~ that 1 he graph of f(:r) = sin .r i.~ con)pre.~~d horizontally by a factor of2 and1hen shifted 5 uni L~ 10 IJle right. (a) \Vha1 is 1 he eql1a1ion for rhe ne\'' graph? (h) \Vha1 i..~ lhe equation i f you tiri;L shi fl by 5 and thencoolpress by 2? (c) m Verif y youranS\\'t -rs by ploningyourequa1ions. 74. Figure 2SshO\\'S IJleg.raphof /(.r) = l.rl + I. ,._,ta1ch IJ'1e function.s (a)-(e) witll ~leu graphs ( i)-(v). (a) ,r = /(x - 1) (b) y = - /(.r) (c) y = - f(x) + 2 (d)y = f(x - 1) - 2 (e)y = f(.< + 1)

SECT ION I . I Real Nu1nbets, Fuoctlons, and Graphs 1t

)' " "

J J '

' ' ' x ,

' -3-2-1, I 2 J -3-2-1 , I

' ' -J-2-1 ,

)'=fl.rl =M+ L (i) ( ii)

)' ) ) 3 3 2 2 2

x x x -3-' ! 1 I ' J -3-2-1 , I 2 3 -3-2-1 , I 2 3

- 2 - 2 -3 -J -J

{iii) { iv) (Y) RGUR 28

75. Sketch ~le graph of y = f (2x) and y = f{!x), where f(.r) = !.rl + I (Pigure 28). 76. Find IJle func tion f \\ho..1;e graph i..~ ob1air1ed J)y shifting tJh~ parabola.\' = .r1 by 3 units 10 IJle riglu and 4 uniL~ dov.n, as in Fig~ ure 29.

...

.. )'= r / , l

' ' = /(x) -4 -----

FIGURE 29

77. Define f(.r) 10 be IJ)e Larger of .rand 2 - .r. Sketch IJle graph of f . \Vhat are it~ doou in and range? Express f (.r) in 1eros or rhe al)solu1e value function.

78. For each curve in Figure 30, state \\hetJler it i..~ syo)O'let1ic \..,ith respect 10 1 he )'

12 C HA PTER l PRECALCULUS REVIEW

80. Suppose that f and gate bo1.h odd. -\\' hich or rhe follO\\ing func-tions are even? \Vhich Me odd? (a) J = /(x)g(x)

(c) y = /(x) - g(x)

(b) ,I' = /(x)3

( I) /(x) ( \' = -- g(x)

Further Insights and C/1al/enges 82. Prove 1.he lriang.le inequali1 y (la + 111 ~ !t.11 + lhl) by adding 1he '"'O inequalities

83. SllO\v Lha1 a fraction r = a / b in lov.ie..c:t 1eritlS hac: afiuite deciou l e:ect 10 tJ)e point (a. 0) 01~ the .ril.(i..~ .

1.2 Linear and Quadratic Functions

J'

\ ' ------ 2

' '1 --- - 1-1 .:.\X 1

b '"' intcrccp1 ~

' ' ' ' . ' ' ' ' '

' '

)'= 1u: + b

_,_,__,_ ______ ...

x I

AGURE 1 T he slope111 i s the ratio " rise o-ver run:

Lii1e..ar f w1ctins 1111ou.sly importanl 1ule in c.alculus. For lhis reason. )'U shuld be thoroughly fan1iliar \vith Lhe basic p1UlJertie.."i of line-

CAUnON Graphs are often plottf!d using differet)t scales for the:r- and )'-11#1S. This is necessaty 10 keep the sizes of grapf)S will)in reasonable bounds. Ho111tJVer, 111heo t /)e scales are different, lines do nOl appear will) their true slopes.

FIGURE 3 Gro .... 1h ofc.

SEC TI ON 1.2 Li1tear a IMI 0.11adla1ic f uoctio1is 13

This foUo\VS from Lhe formula ) r = 111.r + b:

t:,.y ) '2 - YI - =

m(x~ - x i ) =n1

D..t .t2 - .r 1 .t2 - .'I: I

The sk>pe r11 tne..'\sures the rt1te of l'l1ange of y \vith re.;.~pect to .t . Jn f::1ct, by writing

Ci.)' ="' .6..t \Ve see that a 1-unlt increase in .r (i.e., ~.t = I) produce.~ an ntunit c.hange ti.)' in y. For exan1ple, if 111 = 5, I.hen )t incre. 0

14 C HA PTER l PR ECALCULUS REVIEW

AGUR 4 P'.trallel and perpendicul ar lines.

Slope = n1 I ' . SI - I

''IX' - -;;; S lope = 111

-1--------- x - f---------- x tA) Purnllcl line$ lB) PcrpcOOiculnr lines

Next, \'le recall the relatin bet\\i'ee.n tl-.e sk)pe.~ f par tlllel and pe.rpendicular line.'i (Figure 4):

Lines of slope.Ii 1111 ~tnd rn1 are par~tlk'.l if ~tnd only if 1111 = 1r11. Lines of slope.Ii 1111 ~tnd rn1 are pe.rpendiculitr if ~1nd only if

J fft 1 = - -

1112 (or m1m2 = - 1)

CONCEPTUAL ll'ISIGKT The incremenlS over an inlerval [x 1. x1 I: ti..\ = .\"2 - .t1,

are definoo for any function f (linear o r nol), but the rati fly/ llx may depend on the interval (Figure 5). The characteristic pmperty of a linear fonclion f (x ) = mx +b L that ll.y / L\.x has the .sa1ne value 111 for every interval. ln Lher \Vrds, y has a const:1nl rate of change \vi th respect to .t. We can use tl1is prc>perty to test i f l \V q uantities arc related by a line.ar equation.

l!lJ AGURE 5

+--------x Lineiir func tion: th: ra1io (J.y/ Ax is the .snmc over !Ill io1a-wils.

.I'

+ --------- x Nonlinc:ir l'uoction: the n11io .y/ l.\r chrut.t:.-

Real eJpe. arxl pinl pint forms. Given a point P = (a. b) and a slope m . the equation of the line through P with slope 1t1 i.s ) r - }J = nt(.r - a). Similar ly1 lhe line through l\\/O di.stinct ~X>ints P = (a 1. b i ) and Q = (a2. bi) 11'L< slope (Figure 7)

/J2-/J1 tt1 =

01 - l l l

Thcrefc>re, \Ve c..'ln \Vti te its equatin as y - /J1 = 111(:r - 0 1 ) .

Additional Equations for Lines l . Point-slope form of the line through P = (a. b ) with slope m:

I y -b=m(x-a) I 2. Point point form of the line through P = (a1. bi ) and Q = (m. b1) :

y - b 1 = m(x - t11), bi -b , \\/herein= ---a2 -01

18 C HAP TER l PR ECALCULUS REVIEW

)

8

2 P =f). 2)

-l--------""';:-- X 9 12

FIGURE 8 Lin~ thl't>ugh P = (9. 2) \Vilh ?

.slope111 = - j

FIGURE 9 Graphs or (ruadratic fuoction.i.: f(x) = ax2 +bx + ,._

EXAMPLE 2 Une of Given Slope Through a Given Point Find the equation of the line through (9. 2) with slope -}. Slutin ln poinl .. slope fo11n :

2 )' - 2 = -3(.r - 9)

Jn slope interoept foim: .v = -~(x -9) +2or y = -Jx + 8. See Figure 8. EXAMPLE 3 Line Through Two Points Find the cquatin of the line through (2 . I) and (9.5). S olution The line h"' slope

5- I 4 111=-=-9-2 7

Because (2, I ) lies on the line, its equation in point-slope fonn is y- I = ~(.r - 2 ).

A quadr.ilic l'uncti(>n is a fw1ction defir.ed by a quadratic pc>lync>1niaJ f (x ) = llX 1 +bx + ...- (ll, b, c, constanL' wi~l a -F 0)

The graph of f is a par al>0 !ind D = O

No re~ I roots a>OnndD

Cuneiform texts vlfittoo on clay tablets sl>ow that 111e n>ethod of c0tnpleting I/)e square was knov1n to aocien1 Babylonian rnatl)etnaricfal)S wl>o lived s0tne 4000 years ago.

I lgnoru>g air resistance, a basketball follows a parabolic path (figure 10).

FIGURE 10

)'

5

S ECT ION 1.2 Li1tearalMI 0.11adla1icfuoctio1is 11

EXAMPLE 4 Comple ting the Square Complete the square forthe quadratic polyno mial f (x ) = 4x1 - 12.r + 3. Sel'ldicula1'? 4. Suppt)st )' = 3.r + 2. \Vhat i,.l; d ) if .r increa..:;es by 3?

S. \Vhat i.s tlle lll ininl ulllOf /(:r) = (.r + 3)2 - 4? 3. \Vlh~-n i..:; tJle Ii nea.r + by =

18 C HAP TER l PRECALCULUS REVIEW

Exercises In Exen.'ise.i 1-4, find the :t/,111e. the J-inten:ept, and the .t-intert_.epl O.f the line 1vith 1he gien e1p1ati,)11.

I. y = 3x + 12

3. 4x + 9y = J

2. )1=4-- .r

4. y - '3 = 1

(h) j'(:r + h) - /(:r) = 1nh ( for all .r- and h) io1plie..:; 1ha1 f i..:; I inear of slope 111.

35. Find tJh~ rooL.:; of1hequadra1ic 1X>l ynoo\ial.::: (a) /(x) = 4x1 - 3.t - I (b) /(x) = .r 2 - 2.< - I In Eten:i.re.r 3(>-43, t.011111/ete the :tquare anti.find the n1inin1un1 i nL Lhe .\'inl~pL and one other point. 46. If 1he alleles A and 8of1heC)1SLic fibros; i..:; gene occut in a popl1 l ation \\'ilh frequencies I' and I _ ,, ( \\here I' i.'> a frac1ion l>ef\Veen 0 and I} 1 hen Lhe fre(ruency of hetel'OZ_)1gous eanl i s horiioota.I. If a and bare no1 ko,,.vn precisely, "''e 1t1igh1 proceed as follO\\'S. Firs1 bal ance .r b.)' uJ 1 011 rhe left a.:: in (A). T hen S\\.1i1ch place...:; and bal ance .r by w2 on IJle right a..:; in {B) . The a,erage .i = ~(w1 + u12) give..:: an e.~ tin\aLe for .r. Show 1ha1 .r i..; greater IJlan or equal 10 1he Lrue \veigh1 .t" .

b a b

(A) \Bl FIGURE 14

52. Find nun1bers .rand )' \Vith Suri\ 10 aod produc.1 24. flint: Fi11d a quadm1ic potynonl i al satisfied by .r.

53. Find a pail' of nuo\bers \\hose sun) and produC'I are both equal 108. 54. ShO\V 1hat 1he 1')Arabol a .r = .r2 oonsisL.:: of al I ~i nLi; P such 1.hat d1 = d'! , \\'here d1 is the disr.ance frt.lrtt P LO ( 0. ! ) and d1 i..:; IJ)e dis 1ance fr-01t1 P to 1he line). = - } (Figure 15).

) '

ACURE 15

59. Lei a, y oon1pleting 1he square. Lhat IJle parabola ? \' = a.r- + /1.r +

20 C HAP TER l PR ECALCULUS REVIEW

5

FIGURE I The polynool i al .'. = .rs - 5.r 3 + 4x.

I '

5

- l

FIGURE 2 The ta1ional function x + I f (x) = 3 .

.r - 3.r + 2

-''----+-----4-.\ - 2

FIGURE 3 The algebraic function f (x) =Ji + Jx2 - x .

1

Any function tl>al is t>ot algebraic is called transcendemat. Exponent/al and crigooo1nevic functions {J(f! exa1nples, as are the Bessel aad ga1n1na functions Il'lllt appear 11> engu>(!(!fiflg and staristfcs. The renn "zranscenden1at"' goes baek to t11e 1670s. w11en it was used by Gottfried Will>elm Leibniz (1646- 1116) lo describe functions of this type.

and general as they are. apply only l function.~ that a.re sufficiently " \vellbe haved" (\ve \Vill see \vhat \'.:ell behaved means \Vhen \Ve study the derivative in Chapter 3) . Fortunately~ s uch function.~ ~ue adequate for a vast range of application.n1ial i.s ;1Lc;o a rnlional functin fwid1 Q (.t) = JI. An alge.l>raic f'u.octin i..s produced by t1king sutns, products, and q uotients of root.f of polynr exa1nple, Llleequatin y4 + 2.r2 ) ' + J:4 = I define.s y itn plicitly a..~ a fu1lClion of .r.

Exponential func.tion s: The function / (x ) = b' , where/J > 0, is called the e xpo-nential function \Vith ba.se /J . Sme exa1n ple.s ~1re

f (x ) =2' , g(1 ) = JO', (I)' h(x) = 3' p (t ) = (JS)'

Exa1nple l sl'Jows that tf)e cornposition of functions is not eetnmuta1ive: Tl'!f! func110ns fog and go .r 1nay be (and usually are) diff{!(enl.

I lnWJtSe functions are discussed in SectiOf) 1.5.

S ECT ION l.3 The Basic Classes orfuoctio1is 21

Exponential funcLins and their ;,,verses, the lducl, and quotient furtetions:

Cfg)(x\ = f(x)g(x). (!) f(x) - (x ) = -g g\x) (where g(x) fc 0) For example, if f(x) = x1 and g(x) = sin .r. then

(f+g)(x )=x2 +sinx, (f-g)(x)= x 2 -sinx

(fg)(x) = x 1 sinx, (!) (x) = _i__ g s1nx \'4/e can also n1ultipty functiotlS by cnsta.nts. A funclion of ~le frtn

h (x ) = c i f(x)+cig\x) (c,.qconstants) is called a linetr Cirutti

22 C HAPTER l PR ECALCULUS REVIEW

j '= :r+ t

J= I -----.f------x

x< O x ~O

FIGURE 4 A Fune.Lion defined piect\\'ise.

1.3 EXERCISES

Preliminary Questions

EXAMPLE 2 Given the function/, dete1mine i tsdom~1in, range. and \Vhelher or nt it is increasing or decreasing for different value.'> of.'( .

.f(x) = {.~ + 1 \Vhen .r < 0 \Vhen .r > 0 SlutiC>n The function f appe- all .r > I. He1lCe. Ll1e range of the f unctton is {.r : .:r ;::: I}. The functin is neither (ncre.asing nor decrea.'iing for .\' < 0~ ho\vever. Lile function i..'> increa.'iing for x > O.

1.3 SUMMARY For n1 a re.al numbe.r, f (:r) = .r''' is called the JJO\iter fitr1Cti011 with e xponent "' A poly .. nomial P is a su1n of multip~,o; f x'tt. \vhere ,,, is a \Vhole number:

P (.t ) = a11.t11 + t l11- 1.r'1- 1 + +a1x + ao

Thi.~ polyn1ni.al has degre.e 11 (assuming Lhat t111 :fi 0) and tt11 is called the leading coe.f .. ficie.nt. A '" tional function is a quotient P f Q of two polynomials (defined when Q (,t) ;f 0). An algebraic function is produced by L~1king.sum..o;, produclo;,and nlh root.sof polynomins, and inverse higonornetric functions.

A piece\vise-defined f unclin is obtained by defining a function over t\VO o r moredlstinc:t do1n~-tins.

I . Gi ve a11 e.xan,ple of a l'aLional function. for the functions /(:r) = .r 112 and g(x) = - 1 - l.rl ?. 2 .. Is ~v = lxl a polynootial function? \Vhat abou1 )" = lx2 + II? 3. \Vha1 is Utfusual about 1hedo1rtain oft he 0001pooi1e func.1ion fog

Exercises /11 Exen.'ise.i 1-12, de1en11ine the d'1r11aiJ1 of the jiu1,:tio11. I. f(.-' 10. F(.t) = sin ( - , - ) " + I

+ - 1 12 !( - __ ._, ~"--.. .

?

17. x-

f(.t) = . r + S11l .r 18 .

f(x) = 2'

2.rl + 3x 3.r - 9.r-112 19. f(x) =

9 - 7.r2 20. f(.r) =

9 - 7:r2

2 1. f(x) = s in(x2) 22. . t f(x) = J-" + I

23. f(x) = .rears.

" + I 40. Fiod all values ofc SllChthat f (.r ) = ., ha.r;donlain R . . r + 2tx + 4

sarisfies S I' = (.r + I). Then apply E:

24 C HAP TER l PRECALCULUS REVIEW

Q Q __...,._---~'- ... o =a ,,,_-_--r----~-- ... ~ = 21t / ' , -+(----cif-0.o-'','-~\_,\ P

0 I

", 2 ' \

I ' I

' ' ' ' ' ' I I

0 I .+--+ ---+' P = Q + --+ -'----+' p \ () I 0 + -----: T-- +i p 0

' I I \ ' ' / ''..... .,.,,,"

-- _ ..

IA) 18) IC) (0)

18 = - i ' 4 ,

Q

D AGURE 1 The radian foea.i;ure 0 of .a (."()\Jll terclockwise rotation i s the lengt h .along Lhe uni1 circle of lhe arc Ll'g degrf!eS. Tl'!e nu1nber 360 has many divisors (360 = 8. 9 5) , and coasequently. 1nany fractionaf parts of tl)e circle can be ex.J,Yessed as an integer numberofdegrees. Forexa1nple, one-fifth of tJ>e circle is 72, l'HO.nintf>s is SW, rhree-eigl>ihs is 135, ere.

Figure I (A) shO\\IS a unit circle \Vilh radius 0 P rotating counte.rclock\vise into radius 0 Q. Tl1e rntlian 111eas111f! of tl1;s rotatiort is tl1e le11gtl1 8 of tl1e cirr:11lar ore traversed b)' P as;, mtt1tes {1110 Q. On a ci.rcle of radius r. Lile arc traversed by a C represe111 tJ1e sarne a11 gle if lite correspo11tli11g mrt11im1s d{tfer /J}' an i111eger 1111rltiple of 2rr. Por e.xample, rr /4, 9rr / 4, and - J 5rr /4 all represenl the same angle because they differ by muhiples of2rr:

rr 9rr 15rr -=--2ir=--+4rr 4 4 4

Every angle ha.5 a unique radian measure satisfying 0 Se < 21l. With Ll1is choice, lhe angle 8 subte11Cls an arc of length 6 r on a circle of radiu.s r (Figure 2).

Degree .. s are defined by dividing the circle (110l necess:1rily t.l1e w1il circle) into 360 equal parts. /\degree i.s 3ji0 of a circle. A rot.:.ttion Lhrc>ugh 6 degree~~ (denoted 8) is a rot:1tion llu-ough the fraction 8/3t>O of Ll--.e emplel'e cirt'le. For example) a rotatic>n Lhrough 900 is a mta.tion through tlx:. fraction ~.or ~)of a circle.

As \ViLh radians. the degree measure of an angle i.s not unique. Two degree measwe~s represent that same angle if lhey differ by an integer muhiple of 360. For example, the angles -45" and 675 coincide because 15 = -45 + 2(360). Every angle has a unique degree measure 8 \ViLh 0 < 8 < 360.

To e11ve1t bet\1.:een radians and deg1-ces, retnetnbcr lhal 2Jc radians is equal to 360~. Therefore, I radian equals 360/21r or 180/ rr degrees.

To cc>nve1t frrn radians to degrees, 1nultiply by I 80/ JT. To conve1t from degrres Lo radians, muhiply by rr / 1 RO.

EXAMPLE 1 Conver1 (a) 55 to rdians and (b) 0.5radian.s todegree.s.

S olution

(a) 55 x ~ "' 0.9599 radians I 00

(b) 05 radian.' x 1 80" "' 28.648 1T

Co11, e11lion U11less 01J1e1'\vise .t:ttlled, 'e t1fttt1ys tltt!tl.t:Itre t111gle.t i11 rt1ditttL\'.

Hypotenuse <

a

A4jaccn1 FIGURE 3

b Opposit:

FIGURI 4 The uni 1 ci tcle defiili tion of sine and CO..'\i oe is \!al id for au angles O.

f lGURE 5 Four staodard angle..'\: ll)e :r- and } '-COl.)1'di nates of the 1Xli nts are cosO and sin 0.

SECT I ON 1.4 Trigooo1net1ic fuoctio1is 25

The trignometric functions sine and csinecan be defined in te11n.s of rig ht biangle.s. Let 8 be an acute angle in a right triangle, and let us label tbe s ides as in Figure 3. Then

sinB = ~ = oppo.site c hypo tenu.se'

ti adjace-nt cosB= -= ~~~~

c hypotenuse

A dt..s.a.dvanlage f lhe unit circle. Let P = (x. y) be the point on the unit circle corresponding Lo the angle 8 as in Figure,, 4(A) and (B), and define

cosB = .t -coordinateof P. sine = y-coordi11aleof P

Thi..s agree~s. \vi th Lhe righL .. triangle definiti

211 C HA PTER l PR ECALCULUS REVIEW

Gll rtGURE 6 Tiiegraph of y = sin O i s geneiai.ed a.'> the pi0io1 P = (ooslJ, sinU) ll'lOVe.1> around the unit c ircle.

FIGURE 7 Graphs of). = sin O and ,r = cosB O\'et'onepericxl ofleng1.h 2Jt .

11~ often wri1esin .r and cos .r, using .r instead of 0. Depending oa tl)e application, vHJ 1nay think of:r as a11 angle Of sin1ply as a teal 11umber.

FIGURE 8 Si.ne and co..o;ine h.ive period2rr .

Hypotenw;e c

AGURE 9

{/

AdjxcM

I I I I

"

v= t.nnx

b

P =(-~ -fl f lGURE 11

~ I 3i I I

FlGURE 12 co..:; .t" = 1 ror .\ = :l:j

-sin 81 (D AGURE 13 sin 02 = - sin 01 \\'hen 02 = - Bi or01 =Bi + rr.

.\ =s-in4x +sin2x

- t

:t R 3 2

4n Jn 3 2

FIGURE 14 Solutions of s in 4.t +sin 2-x = 0.

) 3

- t

- 3

ID AGURE IS

Co1npreM hprizontnlly by

:1 the-tor of 2

)' 3

- t

- 3

SECT IO N 1.4 Trigooo1net1ic fuoctio1is 21

EXAMPLE 2 Computing Values of Trigonometric Functions Fu1d the values of the six Lrigonomeuic funclions al .l: = 4rr / 3.

Sn The point Pon the unit circle ctTcsi:x>nding tc> the ~lngle.r = 4Jr /3 lte.~ opposite the point with angle rr /3 (Figure 11 ). Thus, we see that (refer to Table 2)

4rr . .

Jr 4rr rr Sin-,- = -Sill-:;-=-,.-.

.> .> -cos-3 =-cs ~ =-?

.> -

The rcniaining value.Ii ~tre

Lan 4rr = sin4rr/ 3 = -JJ/2 = v'3, 3 cos4rr/ 3 - 1/2

4ir cos4 rr/ 3 J3 COL- = = -3 sin4rr/ 3 3

4 rr I I sec - = = -- = -2,

3 cos4rr / 3 - 1 / 2 4rr I -2JJ

csc - - - 3 - sin 4rr/ 3 - 3

EXAMPLE 3 Findtheangle,,xsuchthat seex =2.

St>lutk>n Because sec.r = l /cos.t, \Ve tnll'\t solvecos.r = ~ Fmtn Figure 12 \Ve see that x = rr/3 andx = -rr/3 are solutions. We may add any integer multipleof2rr, so the general solution is .r = .rr i3 + 2rr k for any integer k .

EXAMPLE4 Trigonometric Equation Solvesin4x+sin2x=Ofor x e [0.2rr). Slutk>n We 1nusl find the ang~'\ .r such that .sin4.r = - sin 2.r. First, let's de.termine \Vhen angle .. '\ 81 and 62 .satisfy s ln61 = -sin 8 1. Figure 13 s llO\V.S thal this occurs if 82 = -81 or Bi = 8 1 + rr. Because tl-.e sine function is per iodic \Vi th pertod 2Jr.

\vherek i.~ ~ln integer. T~tking 81 = 4.t and 8 1 = 2x. \VC see that

s in4x =-sin2r 4x =-2r+2rrk or 4x=2r+rr+2rrk

The first equation gives 6.t = 2rrk or .t =(Jr /3)k atld the sec-ond equation give~~ 2x = rr + 2rrk o r x = rr / 2 + rrk. We obtain e ight solutions in [0. 2.-) (Figure 'I 4):

.\' = o. Jr 3' 2 rr 3'

4rr 3'

5rr 3

;md Jr .\' = 2 3rr 2

EXAMPLE 5 Sketc h the g ra ph of f(x) = 3cos (2 (x + ;. )) over[O, 2rr J. St>lutk>n The graph is obtained by sc.aling and shifting the grJph of )1 = cos.t in three ste ps (Figure 15):

Cmpre.~s horCz.nLally by a fa:.to r of2: Shift to the left rr/2 uni Ls:

Shift lcf1 :r/2 tn\it~

{B) y-=cos2:r ( periodic \\'ith period :r)

3

- t

-3

. rt 2

)1 =cos 2x

\' =cos(z(x+ ~)) Ex pond ~-crtica lly by ti ft1t'lor or l 3

- h l--1-+-+- + _,_.r - t

-3

2B C HA PTER l PRECALCULUS REVIEW

CAUTION To SIUJt tl1e graph of). = ('(t:; 2.t to tl)e left ;r / 2 units, we 1nust replace .r by .t + r lo obtain oos (2 (x + ~ )}. It is 111correc1to1akecos('2.r +~) Note thac 10 shift left (111 t11e - .r direction}, we add "Ir/ 2. To s/1ih nght(in tl-:e +.r direction). we subflact rr / 2. counter to what you 1night expect.

The expression (sin .\'")k is usually det1oted siok .r. For exa.7>ple, si n1 .r is 111e squate of si o .r. ~.Ye use sitnilar notation for t/1e other tr1go;1on'Jefl1C functions. H(Nl(Ner, vte teserve sin - 1 .r for tl1e invmse sine function discussed in the l)e.Kt section, rather ttwn for "'!.r.

x 0

~ - x 2 b

FIGURE 16 For>1))pleo~nt.a1')' angles, the .sine of one angle is equal to 1.he oosine of the co1nple1nen1ary angle.

Hypotenuse 5

Adj;1ccm 2 RGURI 17

Opposite .ffi

Expand vertically by a factor of 3:

The verticaJ heighl taken on by .such a function is ll-.e ~tn1plitude. So in Lhi.'i lasl example~ Lhe ainplitude wa .. 'i 3.

Trigonometric Identities .A.. key fea ture of trigo1101n etric functions is Lhal Ll-.ey .satisfy a large nu1n l-,er of idenLities. First and foremosl, sine and cosine saLisfy a funda1nental ide ntity, \Vhich is equivalenl to Lhe Pythagorean Theorem:

I .sin2 .t + cos2 .t = I Equivalent vc.rsions areobt.:1i11ed by divid ing Eq. ( I ) by cos2 x r s in2 x:

I tan2 x + I = sec1 x , Here is a list of some oth er emmonly used identilies. The identities for cotnplementary angles are j ustified by Fig we 16.

Basic Trigonometric Identities

Co n1plernentary angles: .sin ( ; - .t ) = cos.r, cos ( ; - .t) = sin .r

.A..ddition fotmula..~: sin(.r + )') =sin .t cs )~ + cos.t sin y CS(.t + )') = COS.t (.'()S y - .sin .t sin y

Double.;mgle fmmulas: sin2 x =~( I -c,2r), cos2 x =~( I +cos2x) cos 2r = cs2 .r - sin2 .t, sin 2.r = 2.sin x cos.r

Shifi formulas:

EXAMPLE 6 (a)O n Firs t. using lhe identityco.

a

FIGURE 18

b

a

(cos 19, sin(/)

8 ,

FIGURE 19

1.4 EXERCISES

Preliminary Questions

SECT IO N 1.4 Trigooo1net1ic fuoctio1is 29

THEOREM I Law of Cosines If a triangle has sides a. b. and c. a nd() is the angle pposite.side c, then

C1 = a1 + b2 - 'lnb cos()

If 8 = rr / 2~ then cos 8 = () and ~le La\v f C.'iines. reduces to the Pytllagore. multiply by re / 180. lJnless o the.1"lvise slated. all angle.~ in Lhis te xt are give n in radians. T he functions f(x) = co.'() and /(xl =sin () are defined in tenR, o f rig ht triangles for

acute angle.~ and ~-t~ coc>r: (a) I Jr (b) -

3

4. Convert froo1 degrees to radians:

5 (c) 12 (d)

3ir 4

(a) 1 (b) 3

311 C HAP TER l PRECALCULUS REVIEW

6. C31culate I.he \a(ue.1; OftJle SiX SLandard t.rigonoo.etriC functiCO = 2

14. sec t = 2

I S. Fi II in 1.he follO\.\'i ng i.able of values:

" " " "

., 3ir -~ 0 - - - - - -6 4 3 2 3 4 ranO socO

16. Cc.lo1ple1e IJle follo\\'ing table of $igns:

8 sinB oosO tanO 0010

" 0 < B < - + + 2

" 2

33. Use 1headdilion forn\ula 1ooon1pu1e oos (j- + f) exactl y. 34. Use 1.he addiLion forn\ula 10 OOll)J>ure sin ( )- - f) exoc1ly. /11 E.i;ert:i.Te:r 35-Jb', :rkett:h the gro11h 'Jver 10, 2Jr J. 35.

37.

38.

f(O) = 2 sin 40 /(II) = oos (10- ~) f(B) =sin (2 (o- ;) +ir) + 2

39. Deternl ine a func1ion tJlJI \\'Ould ha\fe a ,graph a.~ in Figure 26(A ). staling tJle peiiod and anlplirude.

)' )

-- 1 ---- -----

- - 3 ---- ---- - ----- --

( Al (B) AGURE 26

40. Dete1n,ine a function tJlal \vould have a graph a." in Figure 26(B), Sta Ling 1he period -and anlpli rude. 41. I IO\'.' nlany J)l)illL" lieo111he i n1ersect ion of lhe horizontal lirie JI = c and tJle graph or ,v = sin .T forO :: .T < 2;r '!fl int: The anS\\er d!pends on '-..

42. I IO\V 1ilany point~ lie on IJle i nTerJ:ect ion of the hori'i..Orl tal li11e.r = ,. and tJle graph of)' = ian .T for 0 !: .r < 27''! /u E.r;en:i:re:r 43-46 . . Tolvej()r 0 ~ 0 < 27' (.fee Exa11111Je 4). 43. sin 20 +sin 30 = O 44. sin O = sin 20

Fur t11er Insights and Challenges 60. Use Figure2.R 1o derive Lhe Law of Cosine." fro1n IJle P)lhagorcan Theorenl .

b

0 a

ll - b 0.)S(J AGURE 26

61. Use IJle addition forn)ula 10 r110 \1e oos 39 = 4 (.'()S3 () - 3 (.'OS B

62. Use tJh~ addiLio11 fornlula.~ for sioe and co..;ine 10 prove

( b) 1ana + 1anb l:tll " + = -----1 - tan a tan b

b) Ct)l aoo1b + I OOl (a - = -----COl b - 001 a

63. Let 0 be Lhe angle be1 \veen IJ)t' line )' = 111.r + b a1id IJle .r-aiis rFigure 29(A)J. Prove tJlal 111 = i.an 0 .

SECT ION 1.4 Trigooo1net1ic fuoctio1is 31

45. oos40 + oos 20 = O 46. sin O =co..;20

In E.r;erti:ws 47-56, deri \'e the itlen1i1y us.ing tl.e identities Ii.fled in thlr :te(.ti,)11.

47. c

32 C HAPTER l PRECALCULUS REVIEW

J

cze=:o 1-

FIGURE 1 A Fune.Lion a1~d it iuverse.

In ge1>eraf, 1- (:r) #: 1; ... , . T/'}(! &pression 1- 1 (.r) is sitnply a notation for the inver.;e function, and the '- J does not 1t1prese1>i M e.xpC1>e111.

. .. REMINDER The "domain" is ll>e St!l of numbetS .r such that /(:r) is defined (the sec of a/lav1able fl)puts). and 111e " range" is me se1 of all values /(.r) (ti.,. se1 of outputs).

) '

y=J1(x) = kx+9

" - 18

- 18

AGURE 2

1.5 Inverse Functions f''lt any impo1t anl f uncLion.s,such as logarithn1s, rcx>L'i, and Lh e arc..'iine) are defined as. inverse fu1lC'Lions. In this secLion, \Ve revie\V inverse functlc>ns and their gr~1phs, and \Ve discuss lhe inverse trigonomeL1ic functions.

The inverse off, de no led 1- 1, LS the function that re1erse.nhe effect off (Figure l ). For ex;.1n1ple. LJ1e inverse off (.r) = .r3 is ~te cube root function 1- 1 (.t) = .t 113. Given a table of function v

I ~~otl'~r standard uJrm lot one-to-one is ln/ectlVB.

t lGURE 3 Aone-10-one function i.akeson each ''alue a1 Ol0$1 once.

Thi1>k of a function as a device t0r .. labeling" rn(!lnbers of 111e raage by n1etnbetS of the domain. Wlum f is one-t-one. this labefirg 1s unique and 1- maps eacl> nun1blff in tl1e range back 10 its label.

SECT ION 1.5 lmtrst fu11etic1is 33

To check ow calculation, lei's verify Lhal 1 - 1(/(.r)) = x and /(/-1 (x)) = x: I 1-1(/(.r)) =/- 1{2.r- 18) = -(2.r- 18)+9=(.r-9)+9=.r 2

Bec..iuse 1- 1 is a linear functin, its domain and range \vever. -some functions do not have

an inverse. Consider f(:r) = .r'2. Wl-.en \Ve interchange the colu1nns- in a table f values (which should give us a table of value,, for J- 1). the resulting table does not defi ne a function:

Fu1lCtion Inverse ('!) .f f(x) =x2 x ,-1 (.r)

- 2 4 l.ntctchtl.ngc colwnns) 4 - 2 f-1(1) ha.< !WO =

- 1 I I - 1 ' 'a lue...:;: I and - 1. 0 0 0 0

2 4 4 2

The pmbien1 is lllsiLive nun1ber oocurs l \Vice as -ne. \vhich me.ans that f Lakes on each vnlue at rnSl once. (Pigure 3). Here is the frmal definiLin:

DEFINITION One-to-One Function A function f LS one -Lo-one on a domain D if, for every value c, the equution f (.r ) = c has al 1nst 01ie .slutin fr .t D. Or> equivalently, if for all a, b D,

o

I

31 C HA PTER l PRECALCULUS REVIEW

FIGURE 4 In passiJ1g fro1tl f 101-1, 1.he douu i n and range are in1e1'C-hanged.

-=-=-=-:.-::.-:;,-M+, :.-:.-:.-:.-:.-:.-.:-.:-.:-.:- x I t I j I I I I I

3.\' + 2 AGURE S Graph of f(-

)

30

-2

FIGURE 8 Tile increasingfuocLion /(.r) =.rs + 4.f + J S1Hi.sties 1he l lorizontal Line Test.

x 2

Onc-to--one fur x ?; 0 )

' ' ' ' ' \

\ \

\ \

\

2

\ ' ,

-

311 C HA PTER l PRECALCULUS REVIEW

Do aor confuse Il)e inverse sin- 1 .r wfrh the reciprocal

I (sin .r)- 1 = -.- = csc.r s1 n .r

T/ r, . - 1 -I ' ' ){f Jl)V(!(!X! 1t.Jt)Cl!OOS Su~ .r' CO..t; .r . .. . are oltf!t> denoted a resin .r, aroco..; .r, etc.

FIGURE 13

Su1nmary of inti(!tSe relation berwefN1 1 he sine aad atcsine ful)ct1ons:

. ( . _, ) .Sul .Sul .r = .r lbr - 1 !: .r 5 I

sin- 1(.sin O} = IJ '!( tr ll>t - - 5 0 5 -2 2

)

... ---....

, ' I \ -+--,-:T--71

- 1

FIGURE 15

8 J i - ,,

t lGURE 16 Right IJ'iangle OOllS U'llcted SllCh 1ha1sio O = .r.

'

- - --:: 2 y= wn 1 x

~~ -----~1-:::::==~.v flCURE 17

- 1

SECT I ON 1.5

I-- ->,---+- 8

cos8 \\ilh restricted domain

lmtrst fu11etic1is 31

8

-+--1--+-x - I

EXAMPLE 7 Find an alternative form in le1ms of x for each of cos(sin- 1 x ) and tan(sin - I x ). Sn ThL'i probletn ask.s for Lhe values: of c.os8 ~1nd tan8 a l the angle 8 = sin- 1 J:. Consider a tight t1iangle \Vith hypotenuse of le.nglh I and angle 8 such that sin 8 = .r, as in Figure 16. By the Py~1agrean Therem. the adjacc.nl side has length ./ I - .t 2. N\V \Ve can re.ad off the values fr1nelli c ide.nlities. Because sin8 = .\,

We are justified in taking the positive square men in e itllt'r apprc>ach bec~1use e = sin- 1 .r lies in [-~ .. f] a11d cos8 is posilive in lhis inte rval.

We no\v address ~te ren1aining trigonmetric function.~. Tl-.e function /(8) = u1n8 is one-to.one o n (-! f ), a1xl f (8) = col8 is onc-to ... one on (0, ;rr) (~~Figure. 10 in Section 1.4). We define tl"teir inverses by restric ting them l lhesedon1ains:

e = Lan- 1 .tis the unique angle in (- ~, ~) such lhal Lane = .t

9 = cot- 1 x is the unique angle in (0, rr) such !hot cot 9 = x

The range of bo~1 /\IJ) = 1an l1 and .f(l1) =cot 9 is the set of all real numbers R. There fore.8 = tan - 1.\ a1ld 8 =coi- 1.r havedcunain R (Figure 17).

Tlie function f (8) = sec 8 is nol defi 1ied at 0 = ;. , but \Ve see in Figure 18 lhaL it is one lo-

38 C HAPTER l PRECALCULUS REVIEW

FIGURE 18 /(0) = r.ecO is: one.10-oneoll 1.he interval LO. ir I \\1 ill1 ~ reO)Q\fod.

1.5 EXERCISES Preliminary Questions

Figure 18 shows that the range of .f(O) =sec() is the set of real numbers x such thal I.t i ;::_ I. The same is true of .f(O) = cscO. IL follows Lhal both ()= sec-1 x and I) =csc-1xhavedomain {x: ~t i> 1).

-I! 2

' ' ' ' ' ' I

- I

1.5 SUMMARY

ffJ) = Srizontal line intersects the g raph ff in at tnsl one point. T he graph of .r-1 is o btained by reflecting the graph of .f 1hrough lhe line y = x . T he arc.r;11e and t1rccos;ne are defined for - 1 < x < I:

8 = sin- 1 x is the unique angle in [-~ ~) such that .sin 8 = .r 8 = cos- 1 x is the unique ~1ngle in [0, JT] such that CsB = .t

tan - I .t ;1nd cc>t- 1 x are defined fr all .r:

B = tan- 1 .r is the unique angle in (-~ , ~) such that tan 0 = .r () = co1- 1 x is the unique angle in (0. rr) such thal cot () = x

sec- 1 x ;md csc-1 x are defined for fxl ;::_ 1:

(} _ , . lh . le . = sec .t 1s e unique ~mg 1n

8 = csc-1 .r is the unique ang~ in

[ 0. ~) U (; , rr] such th;tt sec()= x [-;,o)u (o, ;J suchtha1 cscO =x

I. \Vhic.h of 1.he follt)\Ying sa1isfy 1-1 (:r) = /(:r) '! 2. T he function f l'llS teenagers in 1.he United States 10 theit l ast nan.es. Explain \\'hy 1.he ill\'e-rse function 1-1 doe.1; nc. exi.i;1. (a) f(x) =x (b) f(xl = 1 - x

(c) f(x) = 1 (d) f(x) = ,fi (e) f(x) = lxl (!) f(x) = x- 1 3. T he follO\.Ving rragnlentof a trainschedule for the Ne\Y Jersey Tran-sit Syste1n d efi1i e_1; a function f fron1 IO\\' llS LO ti 11-.es. Is f one-1.o--0i1e?

What is r-1 (6:27)?

1renton 6:21 l l an1ihon To\vnshit> 6: 27 Prince1on Junction 6:34 Nt\\' BrunS\\'iCk 6:38

4. A hoo1e\..,Otk probleo1 a..; ks for a ske1ch oft he gtaph of the in~ er.re Of /(:r) = .r + COS.r. Frank, af1er lJ')'illg bul failing 10 filld a fOl'fOula

Exercises I . S hO\\' 1ha1 f (:r) = 7:r - 4 is i " "ertible and lind i1s i '~''en.t: . 2. Is f(:r) = .t.2 + 2 one-10-one? If not.

411 C HAPTER l PR ECALCULUS REVIEW

24.

g(x) = . I-' .t 25.

f(x) = 1.-2 '

26.

\Vhen .r < - 1 \ ... he1l .r :.::_ - 1

\Vhen .r < O \ ... her1 .r ~ 0

l.r \ ... hen .r < O x) = ., g( .r- \\'hen .r :=: 0 In 14en:iseJ 27-32. e~a/ua/e 11ithou111.ting a calt:ula1in- 1 !. 2

- 1 2 3(), '"" J3 32. , ;,,- 1 (- 1)

111 E.\en:.'i.re.r 33-12. ' -' 39,8 10.7 l7

Three prope1ties of expone ntial funclionss.hould besingledout fro1n the start (see Figure I fort he cao;e b = 2):

E.r/Jot1e111ialfit11c1i011s are f}(}S;tive: IJ'"' > ()for all .r . The rtmgeof j(x) = b' is the set of all positive real numbe.1,;. f (x ) = I' is incr.,ising if b > I and decreasing if 0 < b < I. Jf b > I. the e xpnential function /(.r) = /J' is not 1nerely incre as ing but is, in a

certain se.nse, rapidly incre asing. Although the te nn "'ra pid incre.ase" is perhaps subjective~ the follov1ing precises1:11e1nen1 is bue: ,f(.r) = b.i: increase .. '\ tnore rapKlly than the po\ve.r function g(x) = x" for all /1 (we will prove this in Section 4.5). For example, Figure 2

TABLE l

.r .rs 3'

I 3 5 3125 243

10 100,000 59.049 15 759,375 14,348,907 ,-_, 9 .76 5.6 25 847.288.609,443

Gordon f\1oore {1929- }. f\

42 C HAP TER l PRECALCULUS REVIEW

A/tl>ougl> Wt'1t1en 1eferences to the nu1nber ir go back 1nore t11an 4000 yeatS. matl>onJaticians first beca1ne aware oi tf1e special role played bye in the sevMteenth century. Tl'K! aotation e v1as inttoduced by Leonhatd Euler, 111110 discwered n1any funda;neatal properties of Ilus impcrta111 nun>JJer.

Jn the next example, \Ve use the fact that f (.\') = IJ" i s 01-.e tc>--one. lo o ther \VC:>rds, if br = /JY, then .r = .v.

EXAMPLE 2 Solvefortheunknown:

(a) 23.ran c.xpo1--.entia.I function is nol }J = J 0 r }J = 2. as one 1nigl1L think al fir.sl, bul rather a ce11ain irr..1lional aun1ber. dcnc>ted bye, \vl10sc v~1lue is approxintately e ;:::; 2.7 18. A caJcula'lr is used to eval1.1d tllal the base is e . . A..nod1er con1mon ntatir el iscxp(x ).

Ho\v ise defined?Thereare many different definitions, but Lhe) all rc.ly on thecaJculus concepl of a limil. We .shall discuss one \vay of deti ning e in Sc.c tion 3. 2. Anc>tl-.er definition is de.~ribed in E.xample4 ofSectic>n I. 7. Fr no\v. \Ve menti graphical desc.1i p ti ns.:

Using Figure 3(A): Among all e.xpc>nenrial functins y = /J.t, b = e is the unique ba>e for which the slope of the tangenl line to the graph al (0, 1) is equal to J. Using Figure 3(B): The numOOr e is the unique nu1nber soch tll

) ' ' I

I

' , I

I I

.\'= e" .,, (0. I ) , -

---

"'--""'-"'-'::-7:'.'=j:::::::::::---x --\ =-~I( ....... (0. - 1) (2. - 1)

. ' \ ---\ Y = -e(.( ?1

I I I

FIGURE 4 Graphing ,\' = - e'.l -2) .

.....

..... 4' ..... ...: i .... _,

~::~ : ..... ! ..l ;--~ -.. ....

i :::i ! ::::;

~ .. , ..i

. .. .

=:: ...... a.-J. " oL .& ' J. ... .1. ...

SeiSo\ograph of 1he 2010 l~lai t i ea1'1.hq1.1ake. \\'hich 1egi..;1ered 7 .Oon 1he Richter sc.ale. 11\e Richter ~ale i.o; ~ed on I.he logaiithn1 (1c.-. ba.~e 10) of 1.he a1npli1ude of sei.-.l'oic \\'a\'eS. Each y,hole-nu1ober increa~ in Richter 1oagni1ude oorre..1:>ponds 10 a tenfold i ncre.ar.e in ao\pl ih1tle and approxioutel y J 1.6 t io-.e-$ ll)Q1'e energ)' < 111~1~ uf.t.~~.w Co1'Ji kNI-; ~/RI ... e natural togaritl'Nn tS denoted 111 .r. Other comn'}()n notations are log.r and Log.r.

FIGURI 5 v = In .r is the i ll\ferse of ,, = e t . . .

SECTION 1.6 Eq>onential and Logaritlunic fuBC1.ia1is 43

EXAMPLE 3 Drnw the g raph of .v = -e(- is obtained by ttansl;.1ting this g raph 2 unlL'i t the

rig~.

Logarithms Log~ulthmic functins are inverses of expc>nential functins. tvlre precisely. if }J > 0 and /J = I , then Lhe logt1ritlm1 to tl1e JJaseJJ, denoted logbx, is lhe inverse of f(.t) =Jr~. By definition. y = k>gh.t if J,Y = .t. -so \Ve have

and logr,(!r') = x

ln other \VOrds. logb.t is lhe nu1nber to \Vl1ich }J musl be raised in rder to get .t. For exan1ple.

1012(8) = 3 because 23 = s 10110(1) =0 because lgarilhm 10 1he bru;e e, de.no led In x, plays a special role and is c;~led the nlllural

logarithm.

\Ve use a calculatr t evaluate logarithnl.'i nun1e1ically. For exan1ple,

In 17 "' 2.8332 1 bec I and negative-for 0 < .r < J. Figures illusttate~'i these fact~ for the base /J = e. Keep in mind that the lgarith1n of a negative nutnber doe~'i noL exist. For ex:1mple, log10(-2) des not exi..'it becalls:e JOY = -2 ha..'i n solu1-ion.

Pr ea.ch la\v of cxpc>nents, there is a corresponding law for k>giui lhms:. T lie rule ,,.,,.,.+Y = 1, :r bY correspcuxls to the rule

ln \Vords: 111e log f)j tt JJmd11c1 i,f tlk! stlltl oftl1e logs. T ve.1ify Ll1is 1ule, ob.se.rve tl1at J}OSb(.r)') = J.'.V = ,,1g;,.r blOb )'

= 1, 1c>g,.,.r+lo8b \'

The exponents logb(xy) and loghx + logb y are equal as ck1imed because f (x) = b, is 0 1lC-.. to-one. The logarithn1 la\\/S are cllected in the f(.>((o\ving Lable.

44 C HA PTER l PRECALCULUS REVIEW

Bnctcria popubtion P 60tXl 50tXl ----------------UltXl JtltXl 20tXl IO(Xl J...-.--

'--ns. Here are so1ne exan1ple.'i:

2

- 3 - 2 - 1 2 - I

_y = sinhx

)

3

2

---+--+---1--~>--+--+-x - 3 - 2 - 1 2 3

"' = cm.h .T FIGURE 8 ~\' = sinh .r is metric identity s in2 .r +cos2 .r = I ha .. 'i a hyperbolic analg:

cosh2 .t - sinh2 .r = I

The addiLion fonnula.'isatisfied by s in .t ~u1d c.S.t alla instead I' the cJrcle: Bec-int (cosh 1, s inh 1) lies on ll--.e hyperbola .r2 - >) = I, just as (cos 1. sin/) lies on the unit circle .r2 + y2 = I (Figure 9) .

.I ) 3

_...-ir-...._ --r - 2 _/ 2 . - -- -- - -~ - ------

.scch.r = ---cosh .r

c.sch .r = ---sinh .Y

.I

_______ J L -----1-----x

- 2 2

y= Ull\h X .\ = C()lh X FIGURE 11 The h)perbolic 1a11genl and 001.ange11t.

2

2

EXAMPLE 6 Verifying the Basic Identity Veiify Eq. (2): cosh2 .r - sinh2 x = I.

Solutkm

cosh .r + sinh .r = e:', csh .r - s inh .\ = e-.t

411 C HAP TER l PRECALCULUS REVIEW

lo verse hyperbolk fuoctlo1lS

F'utk'tion Dooai11

,, =sinh- 1 .r all .r )' =c~h- 1 .r

.\' ~ I )' = 1an1i - 1 x lxl < I )' = C(l(h- I .r lxl > I \' =Sech- I .r 0 We obtain Eq. (2) by multiplying these two equations together:

c..ne c>n ils do1nain and therefore ha,'i a \veil-defined inverse. Tl-.e functions y = cosh.r and )' = sech .r are one to-01ie n ll--.e res11icted dmain ~.r : .r 2: 0 }. We lel y = cn'ih- 1 .r and )' = sech-1 .t deoote d1e co1Tespcu1Cling inverse.'i.

Einstein's Law of Velocity Addition The inverse hyperbolic tangent pb1ys a role in the Special Theory of Relalivity, developed by Albeit Einstein in J 905. One co11'iequenceof this theory is tha.L no object can travelfasLer than lhespeed flight, c ~ 3 x 108 nlfs. Ei11stein real fled that 1J1iscontradici.s a law stated by Galileo n10re than250 yearse.arlier, nan1ely Lhat\~etoc;ries add. l1nagi1-.ea train traveling al tt = 50 nlfs and a man \Valking do\vn d1e aisle in I.he tr~rin at l' = 2 mis. According Le> Galileo, the nk-ln's velocity relative to d1e grou1lCI is tt + u = 52 nlfs. Thls agree.'i \Vil11 our everyday experience. But tlO\V i1nagine an (uruealistic) 1uckel traveling a\vay fmm tl1e eatth al u = 2 x 108 nlfs, and suppose tha.l the rocket fire.."'i a 1nissile \Vith vek>city v = 15 x 10& nlfs (rc.lative to the rocket). Jf Gallle's La\v \Vereconect1 tl-.e velocity of the 1nis.sile relative to thee;.uth \VOUld bett + v = 3.5 x 108 nlls, \vhicb exceed'i Einstein's n1axi1num speed limit of'-.~ 3 x 108 nlls.

Ho\vever, Ein.'itein's l11ery replaces Galileo's La\v \Vilh a ne\V la\v stating I.hat I.he ;t1\1erse l1}7Jerbo/;c ta11ge11ts o.f rielocities add. f\.1ore p1uisely1 if u is l11e rocket's velocity relative to Ll-.eearll1 and u is the missile's velocity rel~1Live to d1e rocket, I.hen Lhe velocity of the missile relati-ve t tile earth (Figure 12) i.'i w~ where

EXAMPLE 7 A rocket trJvel' away from the earth al a velociiy of 2 x to& mis. A n1isslle is fired from the mc.ket at a velocity f 1.5 x I

1.6 EXERCISES Pre/ iminary Questions

SECTION 1.6 Eq>onential anct Logaritlu11icfuBt1.ia1is 41

1.6 SUMMARY / (x) = b' LS thee.\pone111ialf1111c1ion with baseb (whereb > 0 and b '# I). /(.t) = b' is inc~'t.'5ing if b > J ~u1d decre.asing if b < J. lmporta.ntexponent la\vs:

(i) b' bY = b' +S (ii) ~ = b'- Y (iii) b- ' = -};. (h ') (I')" = l'Y

T he number e "' 2.7 I 8. For b > 0 with b '# J. tre logarithmicf1111ctim1 f(x) =Jog,, xis the inverse of / (x ) =

The1tatr1rt1l logt1ritl1111 is the loga1i th1n \Vith ba.sc e and Li; dented ln.r. e1"" =xfor x > Oandht(e') =x for allx . lmpo11ant lg~trithm lav .. 'S:

(i) logh(xy) = logbx + logb )'

(iii} Jog,,(x") = n logbx The lt,V/JetlJOfic sine tttrd cosf11e:

er: - e-:r sinh x = (odd function),

2 The remaining hyperbolic functions:

sinh .r tanhx = --

co. I f (x) = tanh- 1 x , for lx l < I

f(x) = coth- 1 x, for I.ti > I f(x) = sech- 1 x , forO < x :5 I f(x) =csch- 1 x , forx #0

I . \Vhich of the fol k)\\1 ing jllariable.

3. ei-.: = e"+1

a(ldi1ioo." 6. \Vhat are 1he

.a C HA PTER l

13. In I

IS. log2(25/ 3) 17. lo8f>ll 4

19. k)gg 2 + 1084 2 21. lo~4 4R - los. 12

PRECALCULUS REVIEW

14. log5(54)

16. log2(s513 ) 18. log7(492) 20. log25 30 + k)g25 ~ n . ln (.;;;.,715>

23. ln(e3) + In (e4) 24. log2 j + log2 24 26. g3 logg(2) 25. 7log7(29)

27. \Vri te ar; Lhe na1ural log of a sing.le e;(pression: (a) 2 1n 5 + 3 1n 4 (b) 51n(x112) + 1n(9x)

28. Sol \1e for .t : 111(.,.2 + I) - 3 Ln .r = Lo (2). 111 E.\en:.'i.re.r 29-34. !J(J/\'e /')r the 1111hu11v11. 29. 7,Si = 100 JO. 6e- the )1ear 2000.

Further J11sights and Challenges 50. Sll O\\' 1.ha1 lo&, b logb a = I fo1 all a ,b > 0 such 1ha1 a F I and b ;" I.

. log" .r SJ. Ver1f)' IJlat for all .r. IJh! For1nul aholds. klU .r = - for a. b > lo~b ,

Osuchtha1a ;" l,b ;o! I .

52. (a) U$0C IJle addi1ioo forn\ ul.'L" fol' .! I .

46. Coo\pu1e c~.:;;h .\"and tanh .r. assun) i1lg tha1 sinh .r = 0.8. 47. Prove the adtli1ion fontlula fu cosh .\" g.i\'en by co.sh(.\" + .v) = oosh .\" OO:\h )' + sinh .\" si nh ) .. 48. Use 1.he OOdi1io11 foro)ulas 10 pr0\1e

sinh(2.r) = 2t..osh.Y sinh .r oosh(2.f) = oosh2 .,. + si1th1 .r

49. A traill rt)()ves along a track at vekX"iI)' v. Bionic.a " 'alks down1he ai..:;; le or tlh! Lrai n wiLh \elocity u in IJle di rec1ion or 1he train 's ouxion. Co1npu1e IJle veloci '>' u1of Biooica relative 10 IJle g1t)l.11ld using 1he l a\\'S or boLh Galileo and Einstein i 1l IJle foll O\\i ng ca.~es : (a) v = m oils and 11 = 10 oils. Is your c.alcula1or accurate enOLigh 10 detect 1 he di fference bet\\'l!"en 1he l\VO l a\\'S! (b) v = 107 olls and u = 16 oils.

(b) Use (a) to sho\v 1ha1 Ei ns1ein's La\\' of Velocity Addi 1ion I Eq. (3)J i..o; equivalent 10

w = u +v

UV I + ?

"-

53. Pro\e IJlal every function f can be \\Tiuen as a SUlll /(.\") = /+ (.r) + f - (x) of an even funciion / + (.r) .and an odd function / - (.r ). Express J (.\") = 5,,.\' + 8e- x in 1el'l'OS of rosh.\" and sinh .\". /-lint: ,v = /(.\")+ /(- .\") i s an even funC'lion, and J. = j'(.t) - f(- .t) i..-. an odd function.

1.7 Technology: Calculators and Computers Con1puter technology l1a.s va.stly extended our ability to calculate a1xl visualiz.e matl.1emat .. ical relatiorlships. ln applied settings, co1nputers nre indispensable f(.> r solving con1plex s ysten1.'i of equ~ltions and analyzing d:ita, a.s in \veatl1er prediction and n1cdical i1naglng. MatllCmaticians use computers to study con1plex s b'octures such as the f\1~ulllelbrot Set (Figwcs I and 2). We take advaniage of this technology to explore the ideas of calculus visually a1xl 11ume1ici1Uy.

FIGURE 3 Vie\ving rectangles for 1.he graph or /(.r) = 12 - ' - _,1.

Technology is indispJnsable but also has its limitations. Wl)er> shown t/)e compt.Ner* generaced results of a cornpl& calculatiM. the Nobel priz~1inning p~icist Eugeo{! Wigner (1902-1995) is rupo'1ed 10 have said: "It is nice 10 know that th{! cetnputer understands th{! probl(Nn, but I would lik{! to understaad it 100. "'

10

SECT ION 1.7

FIGURE I COO)puter-generated iouge of the f\,taodelb101 Set, \Vhic.h occurs in I.he n\alhe1na1ical l heory of chaos and frac1.al s. (.~'J t~.'W_.,NU ,'l

50 C HAP TER l PRECALCULUS REVIEW

5 .

' ' l

rec.trrngle in (C). Now we can see clearly thal f ha.' three molS. A further zoom in (D) sho\VS Lhat these rLs are k>eming \.;ould provide their locations \vith greming in on the g raph as in (B), \Ve see that the first pc>.sitive root Iles bet\veen 0 .6 a.nd 0.7 and the .second positive rOL lie.s be t\vee.n 2.4 a.nd 2 .5 . Further zooming sho\Y'S that the first root Ls approximately 0.67 [Figure S(C)-1. Continuing LhL> process, we find that the first two moL' arex "' 0.666 and x "" 2.475.

Since f(x ) = cosx and f (x) = tanx are periodic, the picture repealS il-'t:.lf with pe1iod 2JT . . A.ll Slutions are btalned by adding multiple-.'i of 2.?T to Lhe t\vo solution.'i in (0, 2rr]:

x "" 0 .666 + 2rrk and x "" 2.475 + 2rrk (for any intege r k)

5 . y=,2111 '

" )

. ' .

/1 '

; ; 2.J 25

. . .

_,. {../ ,{ / ,_, 1 ' - 7 0.60.7 ( '' 13 0 3 I i , " I : ' .

>' = ('()S ' o." 0.6 O.M 0.7 - 5 - 5

IA\ 1- 7. t3] x 1- 5, 5] 18 ) 10. 3J x 1- 5. 5J /Cl 10.5. 0.7] x 10.55. 0.85) FIGURE 5 Graphs of). = c..r and .'. = 1.an .r.

CAUnON 11'/)L>I> considenng 11)e graph of a function such as)' = In .r (Figure 6), 11 may appear to approacl> a honZ0t1ta! 8S)1nptote, bu! in fact, it does aot. Fer any given horizontal line, tl1e grap/1 evMtualfy nses above ii. Jn tl1is respect, graphing cafcuf/Jtors and c0tnputergrapl1ing syste1ns must be used judiciously.

.I'

------------~"""""

AGUR 6 )'= In .r.

I - 3.t AGURE 7 Graphs off(.

TABLE l

( I ) " "

I+ -II

10 2.59374 1o2 2.70481 13 2.71692 10' 2.71 815 1o5 2.71 827 1cf 2.71 828

FIGURE 8 Graphs of f(.r ) = ( I+ ; )" .

FIGURE 9 PO\\'eroonsuol ption P (v) a.o; a func1ioll of \lelocity v.

SECTI ON l .7 Teclu'logy:Calculators aitdC01nputers 51

EXAMPLE 4 Investigating the Bellavior of a Function How d oe.' /(11) = (I + l/ 11)" behave for large \Vhole .. nutnber values of 11? Does / (n) tend to infinity a .. 'i /r gets l~1rger? Solulion Firs t, we. make a ttible of values off (11 ) for larger and k'rger value.' o f 11 . Table I suggesL' that f (11) doe,, not te nd to infinity. Rather, "'' 11 g rows large r, f (t1) appears to get clser t some value ne;_tr 2.7 18 (a nu1nber resembling e). This is an exaanpleoflimiting behavior that \Ve \Vill discus..~ in Chapler 2. Next, replace 11 by Lhe vatiable x a11d plI the function / (x ) = ( I + l/ x )-'. The graphs in Fig ure8confirm that / (x ) approaches a limit of approxirnately2.7 . We will prove that f (!1) approaches e asn tends to infinily in Sec tion 5.9 .

3 3

2.7 2.1

I I

0 5 10 0 5(() UXlO () 0

IA) Ill, 10) x 10. 31 ( B) Ill UXlOJ x 10. 3)

EXAMPLE 5 Bird Flieht: Finding a Minimum Graphically According to one mo del o f bird flight., the po\ver ccity Lhat minimizes po\ver ensumptin c1Tespc>nds to the l\vest point on the graph o f P. We plo1 P fi rst in a ~uge viewing rec umgle (Fig we 9 ). This figure reveals the genc.r..11 shape f the graph and shc>\VS that P take..s on a 1nini1num v~1lue for u some where between u = 8 and u = 9. In the viewing recks like a snaight line. ThLc;- illustnttes the k>cal linearity f / al x = I.

52 C HAP TER l PRECALCULUS REVIEW

2 1.2 1.05

-------

12 0 2 0.8 1.2 0.95 1.05 0~ I 1.2 (>.nd'i on the c hoice of viewing rectangle. Experiment \Vi th different vie\ving rectangles until you find ne thatdlspk1ys the information you \Vant. Keep in n1ind that the sc.afe..'i along the ~txe.'i may change as you vary tl--.e vie \ving rectangle. T he folk>\ving ;.1re some \Vays in \vhic-h g rJphing c.afculatrs and cotnpuler alge.bra

system.~ c.an be used in ca lculus:

Visualizing the behavior of a func.tlon - Finding solutions graphK-.a.lly or numeric.ally

Conducting numerical o r graphic.ll experiment..~ Illustrating theoretical ideas (such as local linearity)

1.7 EXERCISES Preliminary Questi ons I . Is there a defini te "-'ti)' or choosing Lhe op1 inlill \1ie\\1ing rectangle.

or is i 1 best Loexpel'iO)Cfll unLi l )'Ou find a vie\ving. rectangle appropria1e 10 I.he proble1n at hand? 2. Desc1i be 1he calculatt'X' screen produced \\'hen IJle fu1lction ,r =

3 + .r2 is ploned \\'i IJl a vie .... ing recta1igle: (a) 1- 1, I] x [0, 2] (b) [0,1 ] x [0,41

Exercises 111e exen:i.Jes i111JU.t.tetti,)n .t/i,,11/d be d'"'e 11.ting a grOfJ/iing ta/t:u/a1or ')r '!

9. Plot the graph of f(.r)= .r/ (4 - .r) in a viC\\1ing rectaogle IJlat clearly di splays Lhe ve11ical and hori20011\I asyrnp1ote~1; .

3. Acoording 10 rhe e\fideoce in E:(runple 4, ii appears tJ1at ,((11) = (I + l/11)11 never takes on a \aloegreatei tJla n 3 fol'n > 0. Does this evidence fltr)~e tlla t /(n} ~ 3 for u > O?

4. I kt\v can a g:raphi ng. calculat~ be used 10 ti nd tJle nl i nil'nun-t value or a fuuction?

JO. I llustrate local lineari t)' ffi' /(."ntJls. Find. 10 IJle neate'..'l;J in1eger 1V, IJle nun:tber of lllOlllhs after 'vhich the acu:>unt \ialue double.1;.

In Exen:i.te.t I .~-ll~. in\ie.uigate the beha~;,,r

19. The graph or /(0) = A oosB + 8 sinB is a sinusoidal \\'a"e ror any conSiant~ A aud 8 . Cc.)nfini) 1.his ror (A . 8) = (I , I). (I , 2), and (3. 4) by plonins / . 2.0. Find the n-,,a:nn iu Figure /. 7. Sketch 1hegraplt.;of .\' = f

54 C HAP TER l PRECALCULUS REVIEW

adoo\ain nan.e \\'hen tJle pri((! i.r; S2pe1' 1tlornh .and IOOOcu:;ioo)C1'S bu)' \\'hen the 1>1ice i s SIOO per nlOntJt de1ernl i.ne Lhe deo\and functi on C. \Vha1 is; IJle decrea..QC in the ou.01 ber of C'llS101r1ers for ever}' SI i11crea.r;e in 1.he oos1 of the don1ain nan1es?

29. Find i:he 1ooti;of f(:r) = .i-4 - 4.r2 and sketch i t..;; graph. On \Vhic.h iJ1ter\tals is f dey shiliing 1he g.taph of r = /( ~.r) to 1he righ1 3b uni1s. Use Lhisobser\'ation 10 sketch

~ie graph of y = Hx - 41. 40. Let h (.t) = cos .r and .g(.r} = .t-1. Cible, or .sra1e 1ha1 no O\atch e.'3b-1J

52. f\1atch each quan1i1y (aKd) \Vilh (i), (ii) . or (ii i) if possible, or sta1e 1ha1 no O\atch e.'

Thi$ 'strange. uum::tor represents limit bchoviot 1ha1 nppc.nrcd firs t in ">cnthe.r mcxtcls S1udicdby m..'1c0)logis1 E. Lorenz. in 196.l. 1.~,. .

C,IM}~~-~" StNJJtttl

2 LIMITS

C alc-ulus is usually divided int t\VO bt~-tnches, differenti.:1l a11d integ1~-tf, partly for his tor ical re.asn.s. The subject gre\v out of effo11s in Lheseventeenlh century to .slve hvo important geometric prblems: finding tangent lines to c urve...; (differential c alculus) and Ctn puting are.as under curve~'\ (integr~1( c alculus). However, calculus is a brad subject \Vith n clear bound..'lries. It include.'\ other topics, such a .. 'i the thery of infinite .series, and it ha .. 'i an extraordinarily \Viele range of applicatiotts. What 1n.ake.'i tile.~ methods and applicatins part f cak:ulll'\ is that they all rely on the corecept of a litnit. We \Vill see througl10ut the text ho\V litniL'i allo\v ll'\ to make cotn putatins and solve pm ble1TL'\ that c annot be solved ll'iing algebra ak)ne.

T his chapter intrduce.'\ the limit concept and sell\ the stage for our .study f the detivative in Chapte r 3. The first sectin, inte.nded a.~ tnotivatin, discusses ho\V limil'i

~-trise in the study f rJ.tesof c hange ~md tangent lines:.

2.1 Limits, Rates of Change, and Tangent Lines Limits ~ire about 110\v a f unctton f bel1aves. as .r app1l >