微积分常用公式及运算法则(上册).pdf

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1、微积分常用公式及运算法则常用三角公式：sin2α=2sinαcosα；cos2α=cos2α−sin2α=2cos2−1=1−2sin2αtan2α=2tanα；sin2α=1−cosα；1−tan2a22α1+cosαα1−cosα结合律(A∪B)∪C=A∪(B∪C),(A∩B)∩C=A∩(B∩C);分配律A∩(B∪C)=(A∩B)∪(A∩C),A∪(B∩C)=(A∪B)∩(A∪C);cos2=；tan2=；2221+cosαtanα=sinα=1−cosα；21+cosαsinα对偶律(A∪B)c=Ac∩Bc,(A∩B)c=Ac∪Bc;sin2α=2tanα；cos2α

2、=1−tan2α；1+tan2α1+tan2αtan2α=2tanα；sin2α+cos2α=11−tan2α1+tan2α=sec2α1+cot2α=csc2α积化和差：sinα⋅cosβ=1sinα+β)+sinα−β2(()cosα⋅sinβ=1sinα+β)−sinα−β2(()sinα⋅sinβ=1cosα−β)−cosα+β2(()cosα⋅cosβ=1cosα+β)+cosα−β2(()和差化积：sinα+sinβ=2sinα+β⋅cosα−β22sinα−sinβ=2cosα+β⋅sinα−β22cosα+cosβ=2cosα+β⋅cosα−β22cosα+cos

3、β=−2sinα+β⋅sinα−β22集合的并、交、余运算律：交换律A∪B=B∪A,A∩B=B∩A;初等函数：双曲正弦、余弦、正切及运算y=sinhx=ex−e−x(−∞1)，2y=tanhx=sinhx=ex−e−x(−1

4、oshx•coshy+sinhx•sinhy,sinh(x−y)=sinhx•coshy−coshx•sinhy,cosh(x−y)=coshx•coshy−sinhx•sinhysinh2x=2sinhx•coshx,cosh2x=cosh2x+sinh2x,cosh2x−sinh2x=1.1极限的运算法则：设limf(x)=A,limg(x)=B,那么lim[f(x)±g(x)]=A±B=limf(x)±limg(x)lim[f(x)g(x)]=AB=limf(x)•limg(x)limf(x)=A=limf(x)(其中B≠0)g(x)Blimg(x)设limfi(x)=Ai

5、,i=1,2,⋯,n,那么对ki∈R,i=1,2,⋯n,有lim[k1f1(x)+k2f2(x)+⋯+knfn(x)]=k1A1+k2A2+⋯+knAn,lim[f1(x)f2(x)⋯fn(x)]=A1A2⋯AnP(x),Q(x)为多项式,当Q(x)≠0,有P(x)limP(x)P(x0)lim=x→x0=Q(x)limQ(x)Q(x0)x→x0x→x0对有理分式函数在无穷大处的极限，有当a0,b0≠0时,a0⋯当m=nbm+axm−1+⋯+a0axmlim01=0⋯⋯当mn设limf(u)=A,limu(x)=u0,且u(x

6、)≠u0u→u0x→x0则limf[u(x)]=limf(u)=Ax→x0u→u0重要极限：sinxπlim=1sinx0,a≠1).函数连续性：limf(x)=f(x0)x→x0导数定义：f′(x)=limy=limf(x+x)−f(

7、x)xxx→0x→0f′(x0)=f′(x)

8、x=x0求导公式：(C)′=0,(xµ)′=µxµ−1,(ax)′=axlna(ex)′=ex(lnx)′=1x1(logax)′=xlna(sinx)′=cosx(cosx)′=−sinx(tanx)′=sec2x(cotx)′=−csc2x(secx)′=secxitanx(cscx)′=−cscxicotx1(arcsinx)′=1−x21(arccosx)′=−1−x21(arctanx)′=1+x21(arccotx)′=−