calculus concepts and contexts 2nd ed - james stewart

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1、APreviewofCalculusCalculusisfundamentallydifferentfromthemathe-usefultohaveanoverviewofthesubjectbeforematicsthatyouhavestudiedpreviously.Calculusisbeginningitsintensivestudy.Herewegiveaglimpselessstaticandmoredynamic.Itisconcernedwithofsomeofthemainideasofcalculusbyshowingchangeandmo

2、tion;itdealswithquantitiesthathowtheconceptofalimitariseswhenweattempttoapproachotherquantities.Forthatreasonitmaybesolveavarietyofproblems.TheAreaProblemA¡Theoriginsofcalculusgobackatleast2500yearstotheancientGreeks,whofoundareasusingthe“methodofexhaustion.”Theyknewhowtofindtheareaofa

3、nypoly-AA∞A™gonbydividingitintotrianglesasinFigure1andaddingtheareasofthesetriangles.A¢Itisamuchmoredifficultproblemtofindtheareaofacurvedfigure.TheGreekA£methodofexhaustionwastoinscribepolygonsinthefigureandcircumscribepoly-gonsaboutthefigureandthenletthenumberofsidesofthepolygonsincrease

4、.A=A¡+A™+A£+A¢+A∞Figure2illustratesthisprocessforthespecialcaseofacirclewithinscribedregularFIGURE1polygons.A£A¢A∞AßA¶A¡™FIGURE2LetAnbetheareaoftheinscribedpolygonwithsides.Asincreases,itappearsnnthatAnbecomescloserandclosertotheareaofthecircle.WesaythattheareaoftheThePreviewMod

5、uleisanumeri-circleisthelimitoftheareasoftheinscribedpolygons,andwewritecalandpictorialinvestigationoftheapproximationoftheareaofacircleA
limAnnlbyinscribedandcircumscribedpolygons.TheGreeksthemselvesdidnotuselimitsexplicitly.However,byindirectreasoning,Eudoxus(fifthcenturyB.C.)usedex

6、haustiontoprovethefamiliarformulaforthearea2ofacircle:A

r.WewilluseasimilarideainChapter5tofindareasofregionsofthetypeshowninFigure3.WewillapproximatethedesiredareaAbyareasofrectangles(asinFigure4),letthewidthoftherectanglesdecrease,andthencalculateasthelimitofAthesesumsofareasofrecta

7、ngles.yyyy(1, 1)(1,1)(1,1)(1,1)y=≈A01x01131x01x011x424nFIGURE3FIGURE434
APREVIEWOFCALCULUSIsitpossibletofillacirclewithrectangles?TheareaproblemisthecentralprobleminthebranchofcalculuscalledintegralTryitforyourself.calculus.ThetechniquesthatwewilldevelopinChapter5forfindingareaswillalso

8、Resou