微積分大一基礎知識經(jīng)典講解

-PAGE.z.Chapter1Functions(函數(shù))1.Definition1)Afunctionfisarulethatassignstoeachelement*inasetAe*actlyoneelement,calledf(*),inasetB.2)ThesetAiscalledthedomain(定義域)ofthefunction.3)Therange(值域)offisthesetofallpossiblevaluesoff(*)as*variesthroughoutthedomain.2.BasicElementaryFunctions(根本初等函數(shù))1)constantfunctionsf(*)=c2)powerfunctions3)e*ponentialfunctionsdomain:Rrange:4)logarithmicfunctionsdomain:range:R5)trigonometricfunctionsf(*)=sin*f(*)=cos*f(*)=tan*f(*)=cot*f(*)=sec*f(*)=csc*6)inversetrigonometricfunctionsdomainrangegraphf(*)=arcsin*orf(*)=arccos*orf(*)=arctan*orRf(*)=arccot*orR3.DefinitionGiventwofunctionsfandg,thecompositefunction(復合函數(shù))isdefinedbyNoteE*ampleIffindeachfunctionanditsdomain.4.DefinitionAnelementaryfunction(初等函數(shù))isconstructedusingcombinations(addition加,subtraction減,multiplication乘,division除)andcompositionstartingwithbasicelementaryfunctions.E*ampleisanelementaryfunction.isanelementaryfunction.1)Polynomial(多項式)Functionswherenisanonnegativeinteger.Theleadingcoefficient(系數(shù))Thedegreeofthepolynomialisn.Inparticular(特別地),TheleadingcoefficientconstantfunctionTheleadingcoefficientlinearfunctionTheleadingcoefficientquadratic(二次)functionTheleadingcoefficientcubic(三次)function2)Rational(有理)FunctionswherePandQarepolynomials.3)RootFunctions4.PiecewiseDefinedFunctions(分段函數(shù))5.6.Properties(性質)1)Symmetry(對稱性)evenfunction:initsdomain.symmetricw.r.t.(withrespectto關于)they-a*is.oddfunction:initsdomain.symmetricabouttheorigin.2)monotonicity(單調性)Afunctionfiscalledincreasingoninterval(區(qū)間)IifItiscalleddecreasingonIif3)boundedness(有界性)4)periodicity(周期性)E*amplef(*)=sin*Chapter2LimitsandContinuity1.DefinitionWewriteandsay"f(*)approaches(tendsto趨向于)Las*tendstoa〞ifwecanmakethevaluesoff(*)arbitrarily(任意地)closetoLbytaking*tobesufficiently(足夠地)closetoa(oneithersideofa)butnotequaltoa.Notemeansthatinfindingthelimitoff(*)as*tendstoa,weneverconsider*=a.Infact,f(*)neednotevenbedefinedwhen*=a.Theonlythingthatmattersishowfisdefinedneara.2.LimitLawsSupposethatcisaconstantandthelimitse*ist.ThenNoteFrom2),wehave3.1)2)Note4.One-SidedLimits1)left-handlimitDefinitionWewriteandsay"f(*)tendstoLas*tendstoafromleft〞ifwecanmakethevaluesoff(*)arbitrarilyclosetoLbytaking*tobesufficientlyclosetoaand*lessthana.2)right-handlimitDefinitionWewriteandsay"f(*)tendstoLas*tendstoafromright〞ifwecanmakethevaluesoff(*)arbitrarilyclosetoLbytaking*tobesufficientlyclosetoaand*greaterthana.5.TheoremSolutionSolution6.Infinitesimals(無窮小量)andinfinities(無窮大量)1)DefinitionWesayf(*)isaninfinitesimalasissomenumberorE*ample1isaninfinitesimalasE*ample2isaninfinitesimalas2)Theoremandg(*)isbounded.NoteE*ample3)Definition