the r book mathematics

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1、7MathematicsYoucandoalotofmathsinR.Hereweconcentrateonthekindsofmathematicsthat?ndmostfrequentapplicationinscienti?cworkandstatisticalmodelling:
functions;
continuousdistributions;
discretedistributions;
matrixalgebra;
calculus;
differentialequations.7.1MathematicalfunctionsForthekindsoffuncti

2、onsyouwillmeetinstatisticalcomputingthereareonlythreemathematicalrulesthatyouneedtolearn:theseareconcernedwithpowers,exponentsandlogarithms.Intheexpressionxbtheexplanatoryvariableisraisedtothepowerb.Inextheexplanatoryvariableappearsasapower–inthisspecialcase,ofe=2.71828,ofwhichxistheexponent.T

3、heinverseofexisthelogarithmofx,denotedbylog(x)–notethatallourlogsaretothebaseeandthat,forus,writinglog(x)isthesameasln(x).Itisalsousefultorememberahandfulofmathematicalfactsthatareusefulforworkingoutbehaviouratthelimits.Wewouldliketoknowwhathappenstoywhenxgetsverylarge(e.g.x→∞)andwhathappensto

4、ywhenxgoesto0(i.e.whattheinterceptis,ifthereisone).Thesearethemostimportantrules:
Anythingtothepowerzerois1:x0=1.
Oneraisedtoanypowerisstill1:1x=1.
In?nityplus1isin?nity:∞+1=∞.
Oneoverin?nity(thereciprocalofin?nity,∞–1)iszero:1=0.∞
Anumber>1raisedtothepowerin?nityisin?nity:1.2∞=∞.
Afraction(e.

5、g.0.99)raisedtothepowerin?nityiszero:0.99∞=0.TheRBook,SecondEdition.MichaelJ.Crawley.?2013JohnWiley&Sons,Ltd.Published2013byJohnWiley&Sons,Ltd.MATHEMATICS259
Negativepowersarereciprocals:x?b=1.xb√
Fractionalpowersareroots:x1/3=3x.
Thebaseofnaturallogarithms,e,is2.71828,soe∞=∞.
Last,butperhapsm

6、ostusefully:e?∞=1=1=0.e∞∞Therearebuilt-infunctionsinRforlogarithmic,probabilityandtrigonometricfunctions(p.17).7.1.1LogarithmicfunctionsThelogarithmicfunctionisgivenbyy=aln(bx).Herethelogarithmistobasee.Theexponentialfunction,inwhichtheresponseyistheantilogarithmofthecontinuousexplanatoryvaria

7、blex,isgivenbyy=aebx.Boththesefunctionsaresmoothfunctions,andtodrawsmoothfunctionsinRyouneedtogenerateaseriesof100ormoreregularlyspacedxvaluesbetweenmin(x)andmax(x):x<-seq(0,10,0.1)InRtheexponentialfunctionisexpandthenaturallogfunction(