微积分大一基础知识经典讲解

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1、Chapter1Functions(函数)1.Definition1)AfunctionfisarulethatassignstoeachelementxinasetAexactlyoneelement,calledf(x),inasetB.2)ThesetAiscalledthedomain(定义域)ofthefunction.3)Therange(值域)offisthesetofallpossiblevaluesoff(x)asxvariesthroughoutthedomain.2.BasicElementaryFunctions(基本初等函数)1

2、)constantfunctionsf(x)=c2)powerfunctions3)exponentialfunctionsdomain:Rrange:4)logarithmicfunctionsdomain:range:R5)trigonometricfunctionsf(x)=sinxf(x)=cosxf(x)=tanxf(x)=cotxf(x)=secxf(x)=cscx6)inversetrigonometricfunctionsdomainrangegraphf(x)=arcsinxorf(x)=arccosxorf(x)=arctanxorR

3、f(x)=arccotxorR3.DefinitionGiventwofunctionsfandg,thecompositefunction(复合函数)isdefinedbyNote7ExampleIffindeachfunctionanditsdomain.4.DefinitionAnelementaryfunction(初等函数)isconstructedusingcombinations(addition加,subtraction减,multiplication乘,division除)andcompositionstartingwithbasice

4、lementaryfunctions.Exampleisanelementaryfunction.isanelementaryfunction.1)Polynomial(多项式)Functionswherenisanonnegativeinteger.Theleadingcoefficient(系数)Thedegreeofthepolynomialisn.Inparticular(特别地),TheleadingcoefficientconstantfunctionTheleadingcoefficientlinearfunctionTheleadingc

5、oefficientquadratic(二次)functionTheleadingcoefficientcubic(三次)function72)Rational(有理)FunctionswherePandQarepolynomials.3)RootFunctions4.PiecewiseDefinedFunctions(分段函数)5.6.Properties(性质)1)Symmetry(对称性)evenfunction:initsdomain.symmetricw.r.t.(withrespectto关于)they-axis.oddfunction:in

6、itsdomain.symmetricabouttheorigin.2)monotonicity(单调性)Afunctionfiscalledincreasingoninterval(区间)IifItiscalleddecreasingonIif3)boundedness(有界性)4)periodicity(周期性)Examplef(x)=sinx7Chapter2LimitsandContinuity1.DefinitionWewriteandsay“f(x)approaches(tendsto趋向于)Lasxtendstoa”ifwecanmaket

7、hevaluesoff(x)arbitrarily(任意地)closetoLbytakingxtobesufficiently(足够地)closetoa(oneithersideofa)butnotequaltoa.Notemeansthatinfindingthelimitoff(x)asxtendstoa,weneverconsiderx=a.Infact,f(x)neednotevenbedefinedwhenx=a.Theonlythingthatmattersishowfisdefinedneara.2.LimitLawsSupposethat

8、cisaconstantandthelimitsexist.ThenNoteFr