全国大学生数学竞赛数学分析辅导150题.pdf

《全国大学生数学竞赛数学分析辅导150题.pdf》由会员分享，可在线阅读，更多相关《全国大学生数学竞赛数学分析辅导150题.pdf（12页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

1、References1 SK9)2 13;1.y3R1Yff3n8N3kn8N.2.exmmI|CX4m0,1K73 00,1?:x1,x2v|x1 x2|0),an+1=c+an,n=1,2,.limnan.5.a1=2,an+1=2an,n=1,2,.limnan.6.Sn=1+12+1n lnn,y:limnSn3.7.P 0,ylimnRn+pn11+x2dx=0.8.f(x)3a,bRiemann,K?(,)a,b,x0(,)f(x)3x=x0?Y.9.lABCS:PnnR,dnR:P.10.ekg,n?:?1+1n?n+e?1+1n?n+.11.kf1(x)=px2+75,f2(x

2、)=rx2+152f1(x),fn(x)=rx2+152fn1(x),f2010(x)=2x).12.f1(x)=f(x),fn(x)=fn1(f(x),x R1.e3n0,fn0(x)=x,KfR1f?R1?N.13.xnk.yxnk4:3e/4m:(1)limn(xn+1 xn)=0.(2)3nlimn+n=0 u?n N,xn+1 xn n.114.xn,kxn 0,n N.0 xn4:.(1)y3xnNefSxknlimn+xkn=0.(2)y3g,Ssn:n N,sn 0)k4l 0,KS?nan?k4l.16.a,b R 0 a 0.-:yn=p0 x0+p1x1+pnxnp0+p

3、1+pn.y:limn+xnlimn+yn limnyn limnxn.18.a,bk4m,a b,f:a,b R,bx a,b,limyxf(y)=A(x)3,A(x)R.yf3a,bk.19.f:0,1 0,1?N,.(1)y3x 0,1f(x)=x.(2)bf(0)=0 x 0,1,f(x)0,0,h (0,k (0,k?f(h)f(k)h+k L?1.yf0(0)3?=?L=limx0f(cx)f(x)x3.23.f:0,+)(0,+)gk.b3 0f00 2f.y(1)limx+f(x)=0,limx+f0(x)=0.(2)x 0,+),f(x)f(0)ex.24.a,b Rm,f:

4、a,b R3?:x a,b)?km.y3mm(,)(a,b)f3(,)Y.25.a R,f:a,+)RY.bf3(a,+),y(1)e limx+f0(x)=+,K limx+f(x)x=+.(2)e limx+f0(x)=0,K limx+f(x)x=0.(3)e limx+f0(x)=l 0,K limx+f(x)x=l.26.3Rfk?,0,1,x R,h R,f(x+h)=f(x)+hf0(x+h).fL.27.a,b R,a b.f:a,b R3(a,b)n+1g,f(k)(a)=f(k)(b)=0(k=0,1,n).y3 (a,b)f()=f(n+1)().28.a,b R,0 a

5、 b.f:a,b RY,3(a,b),qf(a)=f(b)=0,y3 (a,b)f0()=f().d(A).29.a,b R,0 a Af0(c)=0.31.P.(1)ePkk:,y R0,P P0?kk:.3(2)P(x)=xn+b1xn1+bn1x+bnP(x)=(x x1)k1(x x2)k2(x xm)kmk1+km=n.yQ=P+b1P0+bn1P(n1)+bnP(n)?kP:.dd,eP(x)=(x x1)(x xn),KAQ/.32.f:0,1 R+R10f(x)dx 0.b3PZ10P2(x)f(x)dx=0,yP=0.33.f:a,b R,g:a,b RY,T.ylimn+Z

6、baf(x)g(nx)dx=1TZbaf(x)dx ZT0g(x)dx.AO,Olimn+Zbaf(x)|sinnx|dx.34.a,b Rm.PC1a,bk3a,bkY8.(1)f C1a,bf(a)=0,yi.Rbaf2(x)dx(ba)22Rbaf0(x)2dx;ii.Rba|f(x)f0(x)|dx ba2Rbaf0(x)2dx.(2)f C1a,bf(a)=f(b)=0.yi.Rbaf2(x)dx(ba)28Rbaf0(x)2dx;ii.Rba|f(x)f0(x)|dx ba4Rbaf0(x)2dx.35.f:0,1 RY.(1)ylimn+nR10tnf(t)dt=f(1).(2)

7、n N,-an=R10n2?tn tn+1?f(t)dt.ylimn+an=f(1).36.xn=nvuut1+?1n?2#1+?2n?2#?1+?nn?2?,yn=nXk=11ksinkn+1,zn=nXk=1tan21n+k,Olimn+xn,limn+yn,limn+zn.37.yedX:(1)?ln(1+x)lnx?x 1 1lnx(x +).4(2)?arctan(1+x)arctanx?x 1 2x(x +).(3)(1 cosx)x2 x1cosx32x2lnx(x 0+).(4)?ax1+ax2+axnn?1xna1a2ane(12nPni=1ln2ai12n2ln2a1a2a

8、n)x(x 0+),p0 a1 a2 1.Z0dx(2 cosx)2.43.a,b 1,yZx0ln?b cosxa cosx?dx=lnb+b2 1a+a2 1.44.?:x2+8xy+7y2=225,z=0l.45.a,b,F(a,b)=Z0?sinx?ax2+bx?2dx4?.46.(a,b,c,d:f(x)=acosx+bcos2x+ccos3x+dcos4x3x 0?,dL.47.I Rmm,f3IY.yXJ?:x If4:,Kf.548.f:(0,+)RYR+0f2(x)dx.?x 0,Pg(x)=Zx0f(t)dt.yR+0g2(x)x2dx,Z+0g2(x)x2dx 4Z+0f

9、2(x)dx.49.f:R2 RXe:f(x,y)=(0,(x,y)=(0,0),sin?|x|+|y|x2+y2?(x,y)6=(0,0).=,f?50.f,gR2.XJ(x,y)=?f(x,y),g(x,y)0,f(x,y)+g2(x,y),g(x,y)0.y.51.R3m8.V,WkoYV=W=0.eS=V+r2W,K(S)=0.52.Enm,B=(e1,en),C=(1,n)E|.bU Em8,PLlBC=.PL=L(U,R)klUR1wN|m.(1)3Rg.,3RnIO.eA.y?y1,yn?=?x1,xn?g.,3RnIO.eyLAP.(2)?kYf,ydet?2fyiyj?=(d

10、etP)2 det?2fxixj?.53.fR2.Ng(x,y,z)=f(x,y)z.(1)3fx(a,b)6=0:(a,b,c)S,dg(x,y,z)=0 xu(y,z)Px=(y,z),y,z,2yz.(2)/)ePDE:2fxyfx2fx2fy=0.654.f:0,1 0,1 R2Xe:f(x,y)=(x2y2(x2+y2)2,(x,y)(0,1 (0,1,0,x=0y=0.(1)yf(x,y)3xuy,OF(x)=R10f(x,y)dy.yF(x)30,1,OR10F(x)dx.(2)y 0,1,aqK,dd(?55.u=x2 y2,v=2xy,uvL(x,y)(u,v).56.xOy

11、dx2 y2=1,x2 y2=9,xy=2,xy=4(b)u?:=.57.dx2+y2+z2=a2Iz2sin2=?x2+y2?cos2(0 )(3IS)N,|d(JaN.58.Ddx+y=1,x=0,y=0,y:ZZDcos?x yx+y?dxdy=sin12.59.xyz,Aux2+y2+z2 4,x 0,y 0,z 0.60.a,pb,XJl,T7=.61.43S?=.y:dd)NNuT%3=71l.62.eDx2+xy+y2 1.OZZDe(x2+xy+y2)dxdy.63.yun=1,2,kZx0dx1Zx20dx2Zxn0F(xn)dxn=1n!Zx0(x u)nF(u)du.64

12、.A=?3x2 byz?i+(2y+3xz)j+?1 4xyz2?k,ORCAd s,XeCl(0,0,0)(1,1,1):(1)x=t,y=t2,z=t3;(2)g(0,0,0),(0,0,1)(1,1,1)k;(3)l(0,0,0)(1,1,1);65.OZL?excosy+xy2?dx?exsiny+x2y?dy,pL4I=cos2,?0,4?.66.ORL?y2+z2?dx+?z2+x2?dy+?x2+y2?dz,pLR3?,?x2+y2=2pz,ux+vy+wz+h=0.p,u,v,w,h?.P529,16(2)767.OIL?x2y cosx+2xy sinx y2ex?dx+?x

13、2sinx 2yex?dy,L:Sx23+y23=a23,_?.68.Sz=f(x,y)F(x,y,z)=0.XJS?:(x,y,z)?zmY,y:|sec|=q1+z2x+z2y=qF2x+F2y+F2z|Fz|.69.C:x2+y2 x=0,CD.I=ZC?3x5+15xy4?dy 10 x2y3dx?pC?CD,O.70.Ox2+y2a2+z2b2=1(0 b.k.4,u C2()C1?,uu|=0.yZZu?2ux2+2uy2?dxdy 0.78.Cx=f(u,v,w),y=g(u,v,w),z=h(u,v,w)xyzmNuvwm0.pdddyZZZF(x,y,z)dxdydz=ZZZ

14、0G(u,v,w)?(x,y,z)(u,v,w)?dudvdw,G(u,v,w)=F(f(u,v,w),g(u,v,w),h(u,v,w).Q(.79.y4Xn=11n2+1 a.yab+a(a+d)b(b+d)+a(a+d)(a+2d)b(b+d)(b+2d)+?b a d,?b a du.82.?PanxnR,0 R +.n N,-Sn(x)=nXk=0akxk.bx0 Rx06=0.y?PSn(x0)xnR0vR0 min?1,R|x0|?.83.(1)an=en,bn=n2(n N).yf:R R,x 7 f(x)=+Xn=0ancos(bnx),n NCa.(2)yfMac-Laur

15、in?Pf(n)(0)n!xn0.984.Panxn?,R=1.b?Pn|an|2,limx1+Xn=0anxn=A.y?PanP+n=0an=A.85.f(x)2,f(x)=x,x 0.-:yn=1n(x1+x2+xn),zn=xn+(1 )yn.yelimn+zn=0,Klimn+xn=0.d:(1)?yexn yn,Kyn1 yn;exn yn,Kyn1 yn.(2)y limn+yn R,limn+yn R.(3)y limn+yn=limn+yn,dd(.90.f:a,b RYf(a)=f(b)=0.(1)ye-=a,bf1(0)=nD(an,bn),K8bn an|n Dk?,Pc

16、(c 0).(2)yt (0,c),(Et):f(x+t)=f(x),x a,b b?k).(3)y(Ec)?k).91.f3a,bY,(a,b)S,u?x,y a,b,3z0ux,ymf(y)f(x)=f0(z)(y x).yf3a,be.1092.f30,Y,un=0,1,2,kZ0f(t)sinntdt=0,Z0f(t)cosntdt=0,yf=0.93.n Nn 2,I Rmmx,y I,x+y I.PEk3Ing8.M0,Mn,Ef8Xe:F=nf E|x I,|f(x)|M0,?f(n)(x)?Mno.y3n 1M1,Mn1 0f F,x I,?f(k)(x)?Mk(k=1,2,n

17、 1).94.f3(0,+)kYR+0 x2f2(x)dxR+0f0(x)2dx.(1)yR+0f2(x)dx.(2)y:R+0f2(x)dx 2?R+0 x2f2(x)dx R+0f0(x)2dx?12.95.unvnun 0(n +),un+1un=n+an+b,pa,b.(1)y?Pun,OPun.(2)A:O?Pun,pun=1 3(2n 1)2 4(2n+2).96.f(x)=x x,x R.(1)y?P12nf(nx)3(,+):,PF(x)=P+n=0f(nx)2n.(2)FY59!m4.97.Panzn?,1.b?Pn|an|2,limx1+Xn=0anxn=A.y?PanP+n=0an=A.98.f:R2 RXe:(x,y),f(x,y)=+Xk=11k2ek(x2+y2).11(1)f?35.(2)fCa?99.|I=RRD11+y cosxdxdy,pD=?(x,y)|0 x 2,0 y a 1?,Oe:J(a)=Z20ln(1+acosx)cosxdx.100.f3(0,+)kY,0 a b.MR3:,P OM=xi+yj+zk,r=?OM?=px2+y2+z2.g=f r,g=2gx2+2gy2+2gz2.=nM|a?OM?bo.yZZZ(g)dxdydz=4?b2f0(b)a2f0(a)?.12