微积分大一基础知识经典讲解

1Chapter1Functions(函数)1.Definition1)AfunctionfisarulethatassignstoeachelementxinasetAexactlyoneelement,calledf(x),inasetB.2)ThesetAiscalledthedomain(定义域)ofthefunction.3)Therange(值域)offisthesetofallpossiblevaluesoff(x)asxvariesthroughoutthedomain.)()(xgxf:Note1)(,11)(2xxgxxxfExample)()(xgxf2.BasicElementaryFunctions(基本初等函数)1)constantfunctionsf(x)=c2)powerfunctions0,)(axxfa3)exponentialfunctions1,0,)(aaaxfxdomain:Rrange:),0(4)logarithmicfunctions1,0,log)(aaxxfadomain:),0(range:R5)trigonometricfunctionsf(x)=sinxf(x)=cosxf(x)=tanxf(x)=cotxf(x)=secxf(x)=cscx6)inversetrigonometricfunctionsdomainrangegraphf(x)=arcsinxorx1sin]1,1[]2,2[f(x)=arccosxorx1cos]1,1[],0[f(x)=arctanxorx1tanR)2,2(f(x)=arccotxorx1cotR),0(3.DefinitionGiventwofunctionsfandg,thecompositefunction(复合函数)gfisdefinedby))(())((xgfxgfNote)))((())((xhgfxhgf2ExampleIf,2)()(xxgandxxffindeachfunctionanditsdomain.ggdffcfgbgfa))))))(())(()xgfxgfaSolution)2(xf422xx]2,(}2{:domainorxxxxgxfgxfgb2)())(())(()]4,0[:02,0domainxx4)())(())(()xxxfxffxffc)[0,:domainxxgxggxggd22)2())(())(()]2,2[:022,02domainxx4.DefinitionAnelementaryfunction(初等函数)isconstructedusingcombinations(addition加,subtraction减,multiplication乘,division除)andcompositionstartingwithbasicelementaryfunctions.Example)9(cos)(2xxFisanelementaryfunction.)))((()()(cos)(9)(2xhgfxFxxfxxgxxh2sin1log)(xexxfxaExampleisanelementaryfunction.1)Polynomial(多项式)FunctionsRxaxaxaxaxPnnnn0111)(wherenisanonnegativeinteger.Theleadingcoefficient(系数).0naThedegreeofthepolynomialisn.Inparticular(特别地),Theleadingcoefficient.00aconstantfunctionTheleadingcoefficient.01alinearfunctionTheleadingcoefficient.02aquadratic(二次)functionTheleadingcoefficient.03acubic(三次)function32)Rational(有理)Functions}.0)(suchthatis{,)()()(xQxxxQxPxfwherePandQarepolynomials.3)RootFunctions4.PiecewiseDefinedFunctions(分段函数)111)(xifxxifxxfExample5.6.Properties(性质)1)Symmetry(对称性)evenfunction:xxfxf),()(initsdomain.symmetricw.r.t.(withrespectto关于)they-axis.oddfunction:xxfxf),()(initsdomain.symmetricabouttheorigin.2)monotonicity(单调性)Afunctionfiscalledincreasingoninterval(区间)IifIinxxxfxf2121)()(ItiscalleddecreasingonIifIinxxxfxf2121)()(3)boundedness(有界性)belowbounded)(xexfExample1abovebounded)(xexfExample2belowandabovefromboundedsin)(xxfExample34)periodicity(周期性)Examplef(x)=sinx4Chapter2LimitsandContinuity1.DefinitionWewriteLxfax)(limandsay“f(x)approaches(tendsto趋向于)Lasxtendstoa”ifwecanmakethevaluesoff(x)arbitrarily(任意地)closetoLbytakingxtobesufficiently(足够地)closetoa(oneithersideofa)butnotequaltoa.Noteaxmeansthatinfindingthelimitoff(x)asxtendstoa,weneverconsiderx=a.Infact,f(x)neednotevenbedefinedwhenx=a.Theonlythingthatmattersishowfisdefinedneara.2.LimitLawsSupposethatcisaconstantandthelimits)(limand)(limxgxfaxaxexist.Then)(lim)(lim)]()([lim)1xgxfxgxfaxaxax)(lim)(lim)]()([lim)2xgxfxgxfaxaxax0)(lim)(lim)(lim)()(lim)3xgifxgxfxgxfaxaxaxaxNoteFrom2),wehave)(lim)(limxfcxcfaxaxinteger.positiveais,)](lim[)]([limnxfxfnaxnax3.1)2)Note4.One-SidedLimits1)left-handlimit5DefinitionWewriteLxfax)(limandsay“f(x)tendstoLasxtendstoafromleft”ifwecanmakethevaluesoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoaandxlessthana.2)right-handlimitDefinitionWewriteLxfax)(limandsay“f(x)tendstoLasxtendstoafromright”ifwecanmakethevaluesoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoaandxgreaterthana.5.Theorem)(lim)(lim)(limxfLxfLxfaxaxax||limFind0xxExample1Solutionxxx||limFind0Example2Solution6.Infinitesimals(无穷小量)andinfinities(无穷大量)1)Definition0)(limxfxWesayf(x)isaninfinitesimalaswhere,xissomenumberor.Example12200limxxxisaninfinitesimalas.0xExample2xxx101limisaninfinitesimalas.x2)Theorem0)(limxfxandg(x)isbounded.0)()(limxgxfxNoteExample01sinlim0xxx63)Definition)(limxfxWesayf(x)isaninfinityaswhere,xissomenumberor.Example11111lim1xxxisaninfinityas.1xExample222limxxxisaninfinityas.x4)Theorem0)(1lim)(lim)xfxfaxx)(1limatpossiblyexceptnear0)(,0)(lim)xfxfxfbxx13124lim423xxxxExample144213124limxxxxx013322lim22nnnnExample22213322limnnnn32xxxx7812lim23Example3237812limxxxxNotemnifmnifmnifbabxbxbaxaxannmmmmnnnnx0lim011011,0,0andconstantsare),,0(),,,0(where00bamjbniajim,narenonnegativeinteger.Exercises)6(),0(3122lim)1.12banbnann)1(),1(1)1(lim)22babaxxxx)2(),2(21lim)31baxbaxx743143lim)1.222nnnn51)2(5)2(5lim)211nnnnn343131121211lim)3nnn1)1231(lim)4222nnnnn1))1(1321211(lim)5nnn21)1(lim)6nnnn443lim)1.3222xxxx23303)(lim)2xhxhxh343153lim)322xxxxx503020503020532)15()23()32(lim)4xxxx2)12)(11(lim)52xxx0724132lim)653xxxxx42113lim)721xxxx1)1311(lim)831xxx3211lim)931xxx61)31)(21)(1(lim)100xxxxx21))1)(2((lim)11xxxx223)3(3li