高等数学《第六版》电子教案及习题-同济大学

1.11.112312⎧⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩⎩1.1shŭshŭshŭshù123ttababa=bababca≤ba+c≤b+ca≤bc≥0ac≤bc*0+01.1naturalnumberNN={012nN={12nintegerZZ={−n−2−1012nrationalnumberQQ=,ppZqNpqq+⎧⎫⎪⎪⎪⎪∈∈⎨⎬⎪⎪⎪⎪⎩⎭realnumberRR*0R∞δ0(x0−δ,x0+δ)x0δU(x0,δ)U(x0,δ)={x|x0−δxx0+δ}={x||x−x0|δ}x0δx0δU0(x0,δ)U0(x0,δ)={x|x0−δxx0+δ,x≠x0}={x|0|x−x0|δ}={x|(x0−δ,x0)∪(x0,x0+δ)}U(x0)U0(x0)U−(x0,δ)=(x0−δ,x0)x0δU+(x0,δ)=(x0,x0+δ)x0δ1,x01.1{x|axb}{x|axb}=(a,b)=U(2ab+,2ba−)x0=2ab+δ=2ba−x0δf(x)x∈Dfg(x)x∈DgD=Df∩Dg≠∅(fg)(x)=f(x)g(x)x∈D(f·g)(x)=f(x)·g(x)x∈Dfg⎛⎞⎟⎜⎟⎜⎟⎜⎟⎜⎝⎠(x)=()()fxgxx∈D{x|g(x)=0}y=f(u)u∈Dfy∈Zfu=g(x)x∈Dgu∈ZgfgDZ∩=/∅Ffg=()[()],{||,()()}fggfgyFxfgxDxxDgxDZ===∈∩y=f(u)u=g(x)xyuy=arcsin213x−y=f(u)=arcsinuu=g(x)=213x−[1,1],(,)fgDZ=−=−∞+∞[1,1]fgDZ∩=−|213x−|≤1{|12}fgxxD−≤≤=D()()fggxDZ∈∩fgfDgDgZfZfgD•••fgDD1.1y=f(x)[0,1]1f(sinx)2()()()gxfxafxa=++−1()singxx=[0,1]fgDZ∩=0≤sinx≤1D=[2kπ,(2k+1)π]k=0,1,2,…212(),(),gxxagxxa=+=−12[0,1]fgfgDZDZ∩=∩=f(x+a)0≤x+a≤1−a≤x≤1−af(x−a)0≤x−a≤1a≤x≤1+ag(x)=f(x+a)+f(x−a)[,1][,1]Daaaa=−−∩+a≥0[−a,1−a][a,1+a]1−aa0≤a12[,1]Daa=−1−a=aa=12D={12}a0[a,1+a][−a,1−a]1+a−a0a−12[,1]Daa=−+1+a=−aa=−12D={12}|a|12[||,1||]Daa=−|a|=12D={12}g(x+1)f[g(x)]g[f(x)]f(x)=2211xx+−g(x)=11x+g(x+1)=12x+Df◦g={x|11x+≠1}∪{x|x≠−1}≠∪f[g(x)]=22222xxxx++−−Df◦g={x|x≠−2,−1,0}D{x|x≠2}DFD{x|x2,0},Df◦g{x|x2,1,0}1.1Dg◦f={x|2211xx+−≠−1}∪{x|x≠1}g[f(x)]=2212xx−Dg◦f={x|x≠−1,0,1}f(x)=11,1,1xexxe⎧⎪⎪⎪⎨⎪⎪≤⎪⎩g(x)=exf[g(x)]()1fx=D1={x|1eex1}∩{x|−∞x+∞}=(−1,0)∩(−∞,+∞)=(−1,0)[()]1,10fgxx=−()fxx=2{|1}{|}[0,1)(,)[0,1)xDxeexx=≤∩−∞+∞=∩−∞+∞=f[g(x)]=ex0≤x1f[g(x)]=1,10,01xxex⎧−⎪⎪⎨⎪≤⎪⎩f[g(x)]=11,1,1xxxeeeee⎧⎪⎪⎪⎪⎨⎪⎪≤⎪⎪⎩=1,10,01xxex⎧−⎪⎪⎨⎪≤⎪⎩)(xf)(xg|()|yfx=2,[()]yuufx==()max{(),()}Mxfxgx=1()max{(),()}[()()|()()|]2Mxfxgxfxgxfxgx==++−()min{(),()}mxfxgx=1()min{(),()}[()()|()()|]2mxfxgxfxgxfxgx==+−−1.1()[()](()0,()1)gxyfxfxfx=≡/()()ln()[()]gxgxfxfxe=1,0sgn:sgn0,01,0xyxxxx⎧⎪⎪⎪⎪===⎨⎪⎪−⎪⎪⎩||sgnxxx=[]yx=x[3.1]3,[3.8]3,[3]3,[3.2]4.yyyy=======−=−()[]yxxx==−x(3.1)0.1,(3.8)0.8,(3)0,(3.2)0.8yyyy=======−=xR∈[]11xxx+≤+1,()0,xDxx⎧⎪⎪=⎨⎪⎪⎩]10[1,(,,)()0,0,1,ppxpqqqqRxx⎧⎪⎪=⎪⎪=⎨⎪⎪=⎪⎪⎩1y=422x−2y=2ln2lnx23y=arctan(cos21xe−)4y=ln1sin31sin3xx−+5y=sin[sin(sin2x)]6y=5arccos2log(1)ax−7y=4542sinxxx−8y=arctan[ax21x−+ln(x2+2)]9y=sin32sinsinxxx⎛⎞⎟⎜⎟⎜⎟⎜⎟⎜⎛⎞⎟⎜⎟⎟⎜⎜⎟⎟⎜⎟⎜⎟⎜⎟⎟⎜⎜⎟⎝⎠⎝⎠1.11y=4uu=2−x2y=uu=vv=2−x22y=2u2u=lnvv=lnww=x23y=arctanuu=cosvv=eww=−21x4y=lnuu=vv=11ww−+w=sintt=3x5y=sinuu=sinvv=sinww=2x6y=5arccosuu=vv=logaww=x2−17y=42uvw−u=sinxv=x5w=x48y=arctanuu=v+wv=nmn=axm=gg=1−x2w=lnhh=x2+29y=sinuu=3xvv=sinww=2xss=sinxy=sinuu=tvt=x3v=sinww=lsl=x2s=sinxy=xxx++y=f(u)=uu=g(x)=x+xx+f(u)u=g(x)u=x+xx+y=2xy=2xy=uu=2x|x|−|y|≤|x+y|≤|x|+|y|−|x|≤x≤|x|,−|y|≤y≤|y|−(|x|+|y|)≤x+y≤|x|+|y||x+y|≤|x|+|y||x|=|x+y−y|=|(x+y)+(−y)|≤|x+y|+|−y|≤|x+y|+|y|1.1|x|−|y|≤|x+y||x|+|y|≥|x−y|≥|x|−|y||x+y|≤|x|+|y||x−y|=|x+(−y)|≤|x|+|−y|=|x|+|y||x|−|y|≤|x+y||x|−|y|=|x|−|−y|≤|x+(−y)|=|x−y|||x|−|y||≤|x−y||x−y|≥|x|−|y||x−y|=|y−x|≥|y|−|x|=−(|x|−|y|)||x|−|y||≤|x−y|ex≥1+xx=0(1+x1)(1+x2)(1+xn)≥1+x1+x2++xnxixi−1n1(1+x)n≥1+nx(x−1n1x=0)xi=xx1x0x2x1x0x2xba0b2a20bab2a2n!2n−1(n2)|sin|||,xxxR≤∈1123121()()nnnaaaaaaan⋅⋅⋅⋅≤++a1a2a3an12123()nnnaaaaaaan+++⋅⋅⋅⋅≤(ab)12=ab≤1()2ab+ab≤221()2ab+||||mmaa=0a≠m=ann11||||mmaa=0a≠m0am·()nmnmmmmmmmmmnmaaaaaaaa++++⋅⋅⋅⋅==≠nnSn=1()2naan+=na1+(1)2nn−d1Sn=1(1)1naqq−−=11naaqq−−S=11aq−|q|11.1n1+3+5+7++(2n−1)=n212+22+32+42++n2=16n(n+1)(2n+1)1·2+2·3+3·4+4·5++n(n+1)=13n(n+1)(n+2)112⋅+123⋅+134⋅++1(1)nn+=1−11n+n(ab)3=a33a2b+3ab2b3a3b3=(ab)(a2∓ab+b2)an−bn=(a−b)(an−1+an−2b+an−3b2+an−4b3++abn−2+bn−1)n(a+b)n=an+nan−1b+(1)2!nn−an−2b2++(1)[(1)]!nnnkk−−−an−kbk++bn321kk=∑=12+22+32=14302k=∑=2+2+2+2=8321kk=∏=12·22·32=3631(1)kkk=+∏=(1·2)·(2·3)·(3·4)=1441sin(90˚+α)=cosαsin(180˚+α)=−sinαsin(270˚+α)=−cosαsin(90˚−α)=cosαsin(180˚−α)=sinαsin(270˚−α)=−cosαsin(x±y)=sinxcosy±cosxsinycos(x±y)=cosxcosy∓sinxsinytan(x±y)=tantan1tantanxyxy±∓cot(x±y)=cotcot1cotcotxyyx±∓sin2x=2sinxcosxcos2x=cos2x−sin2x=1−2sin2x=2cos2x−1tan2x=22tan1tanxx−cot2x=2cot12cotxx−cscαsecαcotαtanαcosαsinα11.1sinx+siny=2sin()2xy+cos()2xy−sinx−siny=2cos()2xy+sin()2xy−cosx+cosy=2cos()2xy+cos()2xy−cosx−cosy=−2sin()2xy+sin()2xy−tanx±tany=sin()coscosxyxy±cotx±coty=±sin()sinsinxyxy±sinx±cosx=2sin(x±4π)=±2cos(x∓4π)2sinx·cosy=sin(x+y)+sin(x−y)2cosx·siny=sin(x+y)−sin(x−y)2cosx·cosy=cos(x+y)+cos(x−y)−2sinx·siny=cos(x+y)−cos(x−y)arcsin()arcsin,arccos()arccos,[1,1];arctan()arctan,arccot()arccot,(,);sin(arcsin),cos(arccos),[1,1];tan(arctan),cot(arccot),(,);arcsin(sin),[,2xxxπxxxxxπxxxxxxxxxxxxπxxx−=−−=−∈−−=−−=−∈−∞∞==∈−==∈−∞∞=∈−];arctan(tan),(,);222arccos(cos),[0,];arccot(cot),(0,).πππxxxxxxπxxxπ=∈−=∈=∈123111nyxxxxx=++++++=−2s=t3t=1341234:1.1ab(a,b)ab[a,b](a,b)abab[a,b]abab(a,b][a,b)b−a∞[a,+∞)(−∞,b)(−∞,+∞)(,){|}[,]{|}(,]{|}[,){|}[,){|}(,]{|}(,){|}abxaxbabxaxbabxaxbabxaxbaxaxbxxbxx==≤≤=≤=≤+∞=≤−∞=≤−∞+∞=−∞+∞◦∙If(x)=0x∈Df(x)Df(x)=0f(x)≠0x∈Df(x)Df(x)Df(x)≢0x∈Df(x)D1.1y=xµµµµ=0y=1µ=1y=x(−∞,+∞)µ=−1y=1x(−∞,0)∪(0,+∞)µ=12y=x[0,+∞)µ(0,+∞)Dfx∈DfZyfDZy=f(x)x∈Dy∈ZxyDfDfyZZfxyx2+y2=a2⇒y=±22ax−x∈Dyx2+y2=a2y=±22ax−cossinxaθyaθ⎧=⎪⎪⎨⎪=⎪⎩ff(x)f()1.1xx0∈Dfyy0y0|xx=f(x0)f(x)