answer book for calculus 3rd (spivak, 1994)

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1、ANSWERBOOKFORCALCULUSThirdEditionMichaelSpivakPublishorPerish,Inc.HOUSTON,TEXASANSWERBOOKFORCALCULUSThirdEditionCopyright©1984,1994byMichaelSpivakAllrightsreservedManufacturedintheUnitedStatesofAmericaISBN0-914098-90-XCHAPTER11.(ii)(x-y)(x+y)=[x+(-y)](x+y)=x(x+y)+(-y)(x+y)=x(x+y)-[y(x+y)]=x2+xy-[y

2、x+y2=x2+xy-y=x2-y2.(iv)(x--y)(x+xy+y)=x(x+xy+y)-[y(x+y+yB=x3+x2y+xy-[yx+xy2+y3]=x3-y3.(v)n-2,n-1(x-y)(xn-144...4yn-24n-2n--2n-1=x(xn-14n-2n-2n-ip-[y(xn-1...¿n-12n-2n-1=xnyn-22n-1n-[xn-1#•··X=xn_ynUsingthenotationofChapter2,thisproofcanbewrittenasfollows:n-1n-1n-1__(x-y)-xiyn-1-jjn-1-jjyn-1-jj=0j=0

3、j=0n-2n-1=xnj+1yn--i-j_jyn-jnj=0j=1n-2n-2__k+1pn-(k+1)n=xn4j+1yn-1-jk=0(lettingk=j-1)=xn_yn(x"¯IAformalproofrequiressuchascheme,inwhichtheexpression+xn-2,yn-1-·-+xyn-24yn-1)isreplacedbytheinductivelydefinedsymbol2xiy"¯I¯J.j=0Alongthewaywehaveusedseveralothermanipulationswhichcan,ifnecessary,bejust

4、ifiedbyinductivearguments.(ab--1)(cd¯I)(ac)(b-Id¯')3.(iv)===(ac)(bd)-1(by(iii))(a/b)(c/d)=(ac)/(bd).12Chapter1(vi)Ifab¯I=cd¯',then(ab¯I)bd=I)bd,=bc.Conversely,if(cdorad(ad)d¯Ib¯(bc)d¯Ib¯',1.Iad=bc,then=orab¯!=cdIfab=ba¯',-b.=b2,==Conversely,ifathena2sobyProblem1(iii),abora=b,then-b,-1.a/b=b/a=1and

5、ifa=thena/b=b/a=4.(ii)Allx.(iv)x>3orx<1.(vi)x>[--1+Á2orx<[-1-Ã]2.(viii)Allx,sincex2+x+1=[x+(1/2)]2+3/4.(x)x>Äorx<Ã.(xii)x<1.-1.(xiv)x>1orx<-a5.(ii)b-aisinP,so-(-b)isinP.(iv)b-aisinPandcisinP,soc(b-a)=bc-acisinP.(vi)Ifa>1,thena>0,soa2>a-1,bypart(iv).(viii)Ifa=0orc=0,thenac=0,butbd>0,soac

6、sewehaveacb.Butifa=b,thena2=b2,andifa>b>_0,thena2>b2,bypart(ix).6.(a)From05x

7、ion,coveredinthenextchapter).(b)If05x