数学分析试题集锦

June21,20062002 1.(10)lim x→0(sin x1−cos x.2.(10)a≥0x1=√2+xn n=1,2,...limn→∞x n3.(10)f(x)[a,a+α]x∈[a,a+α]f(x+α)−f(x)= 11−x2+arcsin xf′(x).5.(10)u(x,y)u∂2u∂x∂y +∂2ux2+y2dx dy dz,Ωz=x2+y2+z2=az(a>0)8.(10)∞ n=1ln cos1ln(1+x2)2√(2).{n.∂x(4). L(e y+x)dx+(xe y−2y)dy.L O(0,0),A(0,1),B(1,2)O B OAB.√2.(15)f(x)=34.15f (x )[0,1]sup 0<x<1|f ′(x )|<M <+∞.n >1|n −1 i =0f (in− 1f (x )dx |≤Ma n6.(15)θθ(x )=+∞n =−∞e n2xx >07.(15)F (α)=+∞1arctan αxx 2−1dx−∞<α>+∞8.(21)Rrr20041.(630)(1).lim n →−∞(1n +2+...+1f (x ))13sin(y1+n<e −2x ,(0<x <1).(3).+∞e −ax −e −bx(5).e x=1+x+x2n14≤e x+y−2. 5.(12)F(x)= Γf(xyz)dxdydy,fV={(x,y,z)|0≤x≤t,0≤y≤t,0≤z≤t}(t>0), F′(t)=3a+n√2 n(a>0,b>0)(2).limn→∞ 10x n√2 0dx3.(5).F(t)= x2+y2+z2=t2f(x,y,z)dS,f(x,y,z)= x2+y2,z≥ x2+y242.(1020)(1).0<λ<1,limn→∞a n=a(a),lim n→∞(a n+λa n−1+λ2a n−2+...+λn a0)=ap+1p≤1q(∀x>0),x=1(4).Riemann R(x)= 1p0,x=0,[0,1] (5).f(x)[a,b]( b a f(x)dx)·( b a1x2−x1=f′(ξ).4.(12)(1). +∞1sin2xx p+sin x dx p>1125.(14)(1).S(x)[0,1]S(1)=0,{x n S(x)}[0,1](2).f(x)=∞ n=0a n x n,+∞ n=0a n x n x=r+∞ n=0a nn+1r n+1. 10ln(1x=+∞ n=117.(20)(1).(),(2).R n(n>1)R n(n>1)20031.(10)(1).{x n},{y n}n→∞y n≤(11).x n>0,n=1,2,...,limx n=0.n,x→+∞k,x n>x n+k.(12).CC xdy−ydx)−nf(x)](n=1,2,...)(a,b)f′(x).(17).f(x)[a,b]g(x)[a,b]f(x)>0,g(x)>0, [ b a f′′(x)g(x)dx]1πlimn→+∞2f′(x)f(x) (0,1]n(a n5.a n>0,n−1,2,···,limn→∞6.(10) n=1∞n2+n+1.2ξ∈(0,1),f′′(ξ)≥4.8.(10)α>0, +∞0e−(α+x2)sin tdx t∈(0,+∞)9.(10)f(x,y)[a,b]×[c,d]ϕ(x)[a,b]a≤ϕn(x)≤b,ψn(x)[a,b]c≤ψn(x)≤d,F n=f(ϕn(x),ψn(x))[a,b]10.(10)f(x)[0,1]x=1limn→∞ 10x n f(x)dx=f(1).3 i,j=1a ij x i x j≤1,Ω11.(10)A=(a ij)3×3Ω12.(10)(a ij)n R n h(x)= i,j=1na ij x i x j,h(x)n 1x2i=1A13.(10)I= Γ(y2−z2)dx+(z2−x2)dy+(x2−y2)dz,ΓΓx+y+z=3214.(10)u n(x)(n=1,2,···)[a,b] n=1∞u n x0∈[a,b]n=1∞u n(x)[a,b] n=1∞u n[a,b]15.(10)f(x)=x(x∈[0,π))20051.(30)a1+2a2+···+na n(1).limn→∞)x2.x(3).(0,1)(0,+∞)().8(4). D1x2+y2,C:x2+2y2=1,(6).a>0,b>0,(a+1b)b.2.(10)f(x)[a,b] [a,b]f2(x)dx=0,f(x)[a,b]3.(10)f(x)(0,+∞)f(x)(0,+∞)4.(10)f(x,y)= x2y x4+y2,x2+y2>00,x2+y2=0f(x,y)5.f(x)(a,b)∃ξ∈(a,b),f(b)−2f(a+b4f′′(ξ).6.f(x)R∀x∈R,f′′(x)>0,∃x0∈R,f(x0)<0,limx→−∞f′(x)=α<0,limx→+∞f′(x)=β>0,f(x)R7.f(x)g(x)[a,b][a,b]∆:a=x0<x1<···<x n=b,∀ξi,ηi∈[x i,x i+1],i=0,1,,···n−1,lim|△|→0+∞i=0f(ξi)f(ηi)△x i=b a f(x)g(x)dx.8.+∞n=0(−1)n1nf(x13.a n>0,lim infn→+∞ln(1/a n)a n(x−1)2(x+2)=(2).y=arccos(1y),dz=(5).D={(x,y)|x2+y2≤1}, D e x2+y2dx dy=(6).L={(x,y)|x2+y2=1}, L x dy−y dy= 2.(20)()(1).limn→∞x n=0,limn→∞n√sin x)x2M=max0≤x≤a|f′(x)|.5.(17)f(x)R2x L>0∀x,y′,y′′∈R,|f(x,y′)−f(x,y′′)|≤L|y′−y′′|.10f(x,y)R26.(17)I= S f(x,y,z)dS,(a>0)S={(x,y,z)|x2+y2+z2=a2},f(x,y,z)= x2+y2,z≥ x2+y2.7.(17)0<r<1,x∈R.(1)1−r2+1a n+121x2−1sin2n+2cos2n.(3).x y=x2y(1,1)(4).f(x)R g(t)= e t t2f(x)dx g′(t).(5). x2+y2 1|3x+4y|dx dy(6).f(1,1)=1f′x(1,1)=a f′y(1,1)=b,g(x)=f(x,f(x,f(x,y)))g′(1).(7).Sx2b2+z2a2+y2c2=1,x>0,y>0,z>03.(121462)(1).f(x)(a,b)f(x)(a,b)(2).a2n−1a2n= n+1n1f(x)}[a,b] 1.(4).f(x)[a,b]×[c,d]g(y)=maxx∈[a,b]f(x,y)[c,d](5).f(x)[a,+∞)e f(x)=Climx→∞f(x)20011.(1).a1=0,a n=a n−1+3y2)e−y;(3).f(x)∈C[A,B],A<a<b<B,limh→0 b a a(x+h)−f(x)n)(n≥2),limn→∞x n2.g (x )∈C 2(−∞,+∞),g (0)=1,f (x )=g ′(0),x =0;f (x )=g (x )−cos xxdx ;(2).I =Sx 2dydz +y 2dzdx +z 2dxdy ,Sz 2=h 21−x 2x =0S =∞n =0(−1)n1+x n ,k >1,x 1≤0.(1).∞n =0(x n +1−x n )(2).∞n =1(x n +1−x n );7.I (α,β)=+∞e−t 44.(1).I (α,β)D :α2+2α+β2≤−322)ln(1+x );(2).f (x )=x +ln (a −x ),x ∈(−∞,a );(a).f (x )(−∞,a )(b).x1=ln a,x2=ln(a−x),x n+1=f(x n)(n=2,3,···),limx→∞x n.2.f(x)x=0f(0)=0,limx→0f(x)x2;(3).f(x)x=04.f(x)x=0limx→0f(x)n)5.(1). xe x e x−1dx;(2).I= S yzdxdy+zxdydz+xydzdx,S x2+y2=1,z=2−x2−y26.f(x)∈C[a,b],f(x)f(x)f(a)=f(b),(a,b)ξ,f′(ξ)>0.7.F=yz i+az j+xy k x2 b2+z2n)≤e−x(n∈N,n≥x≥0);(2).limx→∞ π0(1−x1+a n;(2).x1=√2+xn,(n=1,2,3···),limn→+∞x n;(3).limn→+∞(1+12.P (1,0)y =√1+x 2dx (a ≥0),I ′(a )I (1);(2).I =Sxdydz +ydzdx +zdxdy2,Sx 2+y 2+z 2=a 2(z ≥0)6.ϕn (x )=(1−x )n ,0≤x ≤1;e nx ,−1≤x ≤0f (x )[−1,1](R )(1).lim n →+∞ϕn (x ),{ϕn (x )}[−1,1](2).limn →+∞ 1−1f (x )ϕn (x )dx ()7.f (x )=+∞n =0=a n x nR =+∞,f (x )=nn =0a k x k ,f (f n (x ))[a,b ]f (f (x )),[a,b ]20041.(20)a 1>0,a 2>0,···,a n >0,f (x )=(a x 1+a x 2+···+a xnx(1)lim x →0f (x )=n√(1)( 10f (x )dx )2≥ 10f 3(x )dx .(2)(1)3.(20)f (x )(a,b)lim x →af (x )=lim x →b,(1)a,b(2)a =−∞,b =+∞(3)ab=+∞ξ∈(a,b )f ′(ξ)=0.4.(20)S (dydzy+dxdya 2+y 2c 2=1(a,b,c >0)20051.(15)limn →∞nk =1sinkn )(n +x n )=elim n →∞x n .3.(15)De −(x +y )2dxdy .Dx +y =1,y =x,x =04.(15)−∞<a <b <c <+∞,f (x )[a,c ]f (x )(a,c )ξ∈(a,c ),f (a )(b −c )(b −a )f (c )2f ′′(ξ)5.(15)x ∈(0,+∞),∞n =0a n x nn =0n !a n+∞0(∞ n =0a n x n e −x )dx =∞ n =0n !a n 20041.(48)(1).lim n →+∞(1a 2+···+nx +1−sin √x 3.(4).∞x =1arctan1(5).1+π49!+π1217!+π815!+···(6).F(x,y)= x x3+x n (n+1,2,···).limn→∞x n3.(15)f(x),g(x)[a,b](a,b)g′(x)=0ξ∈(a,b),f(a)−f(ξ)f′(ξ).4.(14)f(x,y)=xyx2+y2,(x,y)=(0,0),f(x,y)=0(x,y)=(0,0).(0,0)5.(14)I= l ydx+zdy+xdz,l x2+y2+z2=a2(a> 0),x+y+z=0l x6.(14)I= S yzdxdy+zxdydz+xydzdx,S z= h,x2+y2=R2(h,R>0)7.(15)∞ n=1x n(1−x)2[0,1]8.(15) +∞0cos(x2)x n(1,1+δ)3.(15)f(x)= 10|x−y|sin√lnnsin n5.(17)I= Γ(y2−z)dx,Γx2+y2+z2=a2,z≥0x2+y2=2bx,0<2b<azΓ6.(17)f(x)[0,1]min[0,1]f(x)≥− 10|f′(x)|dx.7.(18) +∞0sin xy1−x=3,a,b.2.+∞n=112x+1)n2,3.f(x)[0,1],f(1)=2f(0),∃ξ∈(0,1),(ξ+1)f′(ξ)=f(ξ).4.f(x)[0,1]f(0)=0,0<f′(x)≤1.( 10f(x)dx)2≥ 10f3(x)dx.5.f(x)[a,b]f(a)≥a,f(b)≤b,∃ξ∈[a,b],f(ξ)=ξ.6.O(0,0),A(π,0)L:y=a sin x(a>0) L(1+y3)dx+ (2x+y)dy7.I= S xdydz+ydzdx+dxdy2.S x2b2+z2x+ye−xy dx[0,1]9.x2+y−cos(xy)=0(1).(0,1)y=y(x),y(0)=1.(2).y=y(x)(0,1)(3).y =y (x )(0,1)(4).(0,1)x =x (y ),x (1)=0?20021.(45)(1).(15)ε-δlimx →1(x −2)(x −1)x +1{x n }x 0=1x n +1=f (x n )n =(0,1,2,...)lim n →∞x n =√cos1x 2,(x =0)(4).√n x(1,∞)3.(30)(1).(15)I =x 2+y 2+z 2=R 2dS x 2+y 2+(z −h )2h =R .(2)(15)a,b,ce x =ax 2+bx +c4.(30)f n (x )=cos x +cos 2x +...+cos n x ,(1).(15)n ,f n (x )=1[0,π(2).(15)x n∈[0,1320041.(15)f(x)X f(x)XX{x n}{x m}limn→∞(x n−x m)=0limn→∞(f(x n)−f(x m))=0.2.(15)f(x)(−1,1)f(0)=0,limn→∞f′(x)n)3.(15)f(x)[a,b]f(x)a f′+(a)<0b f′−(b)<0f(a)=f(b)=c.f′(x)(a,b)4.(15)f(x)[a,b]Riemann b a f(x)dx<0[α,β]⊂[a,b]x∈[α,β]f(x)<0.5.(15){Iα}[0,1]δ>0[0,1]x′,x′′|x′−x′′|<0Iβ∈{Iα}6.(15)x=r sinθcosϕ,y=r sinθsinϕ,z=r cosθ∂2u∂y2+∂2u1+cos2xdx.8.(15)u=x2+y2+z2x2b2+z2√√2 ∞0sin x x dx()10.(20)(1)ΩR3∂Ωu,vΩ+∂ΩΩ(u∆v−v∆u)dxdydz= ∂Ω(u∂v∂n)dS∂∂x2+∂2∂z2.(2)P∈R3(ξ,η,ζ),r(x,y,z)=((x−ξ)2)+(x−η)2)+(y−ζ)2).∆14πδ2 ∂B(P,δ)u dS.20061.(20)(1).x n=1+13+...+1n+1+12n.2.(15)f(x)[a,b]r k xlimk→∞f(x+r k)+f(x−r k)−2f(x)∂x(x,y),∂f∂x,∂f(ax2+by2+cz2)31−2x−x2∞ n=0n!9.(15)f (x )f (0)=0x|f ′(x )|≤Af (x )[0,∞)f (x )=0.20031.(16)f (x )=ax −ln x(0,+∞)ax =ln x2.(16)S(x −y )dxdy +x (y −z )dydz ,SV :x 2+y 2≤10≤z ≤33.(16)f (t )[0,1]u (x,y )=1f (t )|xy −y |dt ,0≤x,y ≤1,∂2u∂y 2.4.(16)limx →0x 20(√1+t )t +1x 2(e x 2−1)2.5.(16)(1).∞n =012n +x ,x∈[0,+∞),f (x )[0,+∞)20041.(16)f (x )=x(t −1)(t −3)dt[0,5]2.(16)sin(x +y )+sin(y +z )=1z =z (x,y ),∂2z1+x 2dx +x (y +1)x 2+2n3.(16) S a(1−x2)dydz+8xydzdx−4xzdxdy,s x=e y(0≤y≤a)x4.(16)∞ n=1ln(1+2|x|2n2−1=3x(3)f(x)(−∞,+∞)limn→ 10f(x n1−x.(2)limn→0e x3−1−x3|x|+|y|.3.(168)(1) 2+sin2x(n+1n(ln n)p,(p>0).5.(12)x3−3x+c=0(c)[0,1]6.(12)∞ n=13n+(−2)nx2+y2ds,L x2+y2=ax(a>0).(2)S1n=ab.2.(147)(1)limn→0√1+2x2x)tan x.3.(12)√3(x+1)7.(147)(1)f(ln x)=ln(1+n)1+sin xdx.8(147)(1)∞ n=1(−1)n1+1n(ln n)n.9.(147)(1)f n(x)=x n−x2n,(n=1,2,···),x∈[0,1].(2)∞ n=1x n n,x∈[−1,0].10.(12)u=f(x,y)∂2u∂2y2=0,x=φ(s,t),y+ψ(s,t)∂2u∂2t2=0.φ(s,t),ψ(s,t)∂φ∂t ,∂φ∂t∂φ∂t∂ψ∂t=0.11.(12)I= V z2dxdydz,V x2+y2=1 x2+y2=z2(z>0)z=h(h>1)12.(10)I= L ydx+zdy+xdz,L x2+y2+z2=a2,x+y+z=0, x。