高等数学基础试题及答案.doc.pdf

1)1 y=f(x)2”，/(-x)=/(x),y/(-x)=-/W/(x)3 y=c y=xa(a y=a(aO,aHl)y=logff x(a A 0,a1)sin.r,cosx,tan.r,cotx arcsinx,arccos.v,arctanx 4y y=eu,=v2v=arctan,w=l+x51x+71+x2(x0),f(x)=_ o X x=-,X t 2f(x)=!+Jo ln(x-2)x-2 0ln(x-2)0 x2XH3：5-x0,x5o(2,3)U(3,5 3/(x)0,1/(lnx)/(lnx)0lnx0X-3 0 x-33u(3,JK1 A.f(x)=7?,g(x)=.r；B.f(x)=Inx,g(x)=2In.r:x2 C./(x)=ln.v g(x)=3ln.r：0/(x)=-,g(x)=x-1 解:AF=|X|H XB,DAB,DCC2/(X)/(X)/(-X)A.v=x；BJTC.yW：F(x)=/(x)-/(-x)X F(-x)/(-x)-/(-(F)=f(-V)f(x)=-(/(“)-/(-r)=-F(x)F(x)=/(-x)f(-(-v)=f(-V)f(x)=f(x)01)()A.L：C.(.V _ 1):x*+=x+2+；2=(x+2/(.r+l)=(x+l)2-2 f(x)=x2-2.BD5/(x+-)=.v/(x)=(B2 D1 5:&，/(X)(Y0,F(x)D3/(x)/(x)/(-.v)AB.C.D.B 12 fit:3 lim lim2%X-Xo X-XQ lim+ah hm-x+oo 4.sinx.lim-=1”0 x lim(l+x)x=e)x-0 5(1)Ja2+xk-a lim-lim xQ(J/-a)(4o2 xk+a)1 la lim(l+)x=e bS X lim a JT)TO sin a(x)a(x)=1 lim(Wy”X-1 sin x 3 sinx.1 x Ilin-=lim-5 sin 3x e 3 sin JX 3x sinx lim-EX sin 3x 3 lim-z 3x 2-lim(l+-)22 r x 2(1+-)x lim i(l lim(l+X XX-X X/（X）x=x0v=x0/（X）/（X）x=x0 x=.r060)JS2 1 Q sin 1lim.x=_ o zo sin x,.1 x*sm lim-=lim(xsin-)=limxsin lim X=0 x1=0 sinx z x sinx x sinx limxsin=0 5 X lim=lim-=-=1,lim=1 sin r sinx.sinx 1 x-o x -lim-X XTO x 2/(x)=VSin7*0/(x)x=0/(x).v=0 limxsin-=0 lim(x+l)=l/(O)=l.T-MT X x=0 0)0,+8)/(X)x=0B 3.4.5.6./(X)=X2-3X+2,ffr(x)=_./V)=2x-3,(x)=(2x-3)2-3(2x-3)+2=4X2-1 8X+20 7j=ln(l+x2)1 f(x)=xsinx=0 x A.B.C.D./(x)x=0limxsin=0 x x B2_ sin x A.ex,(x oo):B.-,(.r t x):X C.ln(l+x),(x-1)：D.,(x-0)x IE0 XTR X Ifu A,C,D0B(1r lim(l2)-Jt=lim()x=lim-”x me Y 1 me Y 4 2 3 A 1 X+3(1+-V X x,0-(2.v)s=32x,5 12x,s,r(x-12 x+3)32 8 lim-=12(x-2),s 12 3(4)XTO0.1 v 3x v 1=lim -lim 3 x-0sin3x*TOJ-Y+I 2 xsin+b x 0 x(1)/(x)x=0(2)a,b/(x)x=0/(X)x=0lim f(x)=lim/(x)jrflim f(x)=lim(xsin+6)=h xf x lim f(x)=lim=1 X-M)*XT(T x 6=1lim f(x)=lim f(x)b=ljr=Ox-0-x-M)*1(l)lim-3Y2-I L+4x-12(3)lim(V_1,1O(2A+3)12(x-2)15 2Y+3(2)lim(_i)-E X-l r JX_1(4)lim-x-*sin 3x(x-l)(.r-2)x-1 X2+Ax-12(x-2)(x+6)x+6.x 3x+2.x 1 1:.lim；-=hm-+4.V-I2 x-2 x+6 8 lim(l+-)3 IT*X 1-x-1 sin 3x(V1)xT0sin3x(Jlx+l)=lim JTTOro sin 3x(2)lim/(x)=lim f(x)=f(x0)X-Xo h=f(O)=a,a=b=1x=0 1/(X)x=0)ato Ar F x-x0/(x)A-=.Yof(x)y=/(X)(XJ(xy=f(x)(x0,f(x0y=f(xQ)(x-x0)f(xQ)y=f(x)v=f(x)23F(1)(2)3)(4)(5)31y=-x2-x 2-x 2 r 2 2 y=Vx-2 In y=*In(x+1)-*ln(x-2)y _ _ i _ i 7_2(X+1)3(X-2)_ 0)(k 0(f)4 d(“v)=du dv d(w v)=vd+udv ixiz/-udv d(-)=-;v v*dv=y；dx=yu ”；dx=yfudu J：fl61x=0/(0)=0,/(0)=1,lim JCTO%limhm(0)1 z x kto x-0 1.2V=4=r(1,1)/(X)x=xf(x0)X=1 2 3/(X)=X2-4.V+5,ff,(x)=_ (x)=2x-4 ff(x)=(2x-4):-4(2.v-4)+5=4.r2-24x+37 4x2-24.V+37 1./(v)-./(2)12 x-2A.2x：B.2：C.4；D x y=M)(v H 0)lim2)/(x)=x2,宀 x-2/X2)=2X|T=2=4,C 2/(I)=x fx)=()o X 1 1 1 A.-：B.:C.：D.x x/(X),f(x).x=l/(.r)=f XX X X*X D 3/(x)=(x+l)x(x-l)(x-2)/*(0)=(A.O：B.1；C2 D.-2 fx)=x(x-1)(x-2)+(x+l)(x-l)(x-2)+(x+)x(x2)+(x+l)x(x-l),x=0 0,0)=(0 1)(0-1)(0-2)=2 C4.y=x-ex0A.(OJ)：B.(l,0)：C.(0,-l)：D.(-l,0)y=l-ey=0 x=y(0)=-l,c 5 y=sinx2,y=()A.COSX，；B.-cosx2:C.lxcosx2；D.-2xcosx2 yf=cosx2-(x2)r=2xcosx2 c1 y=tan2x+2Mn,2 y=:+cosx-2MOXln2 cos 2x I 2 71 sin-r rcJ 2/(ex)e/x,/(x)y=/(ex)re/+/(ex)e/(I)r+/(ex)e/(,y.V vdv-xdy ydx+xdv+e dv=-2 x y dx x2yxvey+x 4j（x）t1 X=2 y=1-dLv dy y-1 1|II (lx x 2/2 5.oyo.(TD,O).l+.r 2limEl_x 3y=y(x)fi!.n?+e1 Inydr V 3./(x)=l(ex /(x)=|(ex-e_x),V)=0 x=0 x 0(x)0 f(x)0,x=0/(x)=y(ex+e_t)ZB 1 y=x22,2A)B)C)D)Dy=2x,0fx)00：2,2y=12.y=f(x)c(a4b)b_a A)(a,b)B)C)(a(a”)D)0(aM：D/(x)D3.K fx)=0y=f(x)A)B)C)D)C4(a,b)x0(ayb)f,(xQ)=/*x0)=0.x=x0 DPhD.1y=x-ln(l+x)v=x-ln(l+x)(-L+00).,1 X V=1-=-1+X 1+X y=o.x=o.(-1,0)(o,+ao)-ixo/0：0 .v 0.y=x-ln(l+x)(-1,0)(0,+x)2.108lim In.v lim E(l-x)r=lim x-l 1 A)C)(AO,/(XO)B)D)X.yx2h=10&h=-x j=x2+4xh 2,108 2 432=X+4x=x 4-jr X*,r-432 2432 y=2x+=5 X X 0得2(x3-216)=0=x=6 x6,y0;x6,/lex xe/(x)=lnx,f(x)(0/oc)/(x)l,x1 cx,In x-In 1=(x-1)c ol,1 c lnx x-l eebx e1x1ex xe 5(2)1 2(2,5),o|2.rdv=.x2+c,(2,5)c=l,y=x2+l 2(x)arctan.v2,v)=_ WctaWr 2x r_2(l-hx4)-8.r4 1+x4 _(1+x4)2-3F(x)f(x)/(av+fr)d.Y=_ J/(av+b)dx=j/(ar+b)d(ax)=j/(ar+b)d(ax+b)/V)=(2-6*=d+-r4)2 =f dF(ax+b)=F(ax+b)+c aJ a 1|/(x)dr=xlnx+c f(x)=(A.lnx+1：B.Inx:C.x：D.xlnx ft?：/(X)=(x ln.v)=ln.r+=lnx+1 x A2F(x)/(x)A.(J/(x)dr)=F(x)；C.jF(x)dr=F(x)：B J F(x)(h=/(x)+c：D.(J/(X)CLV)=/(X)-(J f(x)dx)=/(x)|Fx)dx=F(x)+c D3F/(x)Jj/(l-x2)dx=()A.F(1-X2)+C：C.F(1-X2)+C；2 B.-F(I-X2)+C：D.F(x)+c=-”(1-，)(171 C1|XlpLd d(l)X-.X J=-Jd(Jl-x?)=Jl-x2+c x=sin/d.v=cos tdt fVl-Jf2 j fcosr-cos/.pl-sin2/.fz 1.、d/=r-d/=r(-i)dr J xsin*/J sin*r J sin t vi-x2=-cot Z-/+c -+arcsin x+c 2(1)jarcsin.rdv k j arcsin,vdv=x arcsin.r-j.vd(arcsin x)=x arcsin x-j(*dx=x arcsin.v+f/d(l-x2)J2Vl-.x2 =Inxd+J d(ln.v)Inx r 1,lnx 1 -+dr=-+c X J X*X X （1）.(2).A-f(x)dx=0 C f x2dx x dx Jo Jo B.ch&J gdx=|jxdx(3x2dx3t2dt A.(cosM)=cosx B.=cos.v C.（cosfd=0 D./（x）-04 L:BD.f(x)dx=()o B.0 c.21 f(x)dx j D.f fx)dx g(x)a,bx=a,x=bA.|/(x)-g(x)|B.(/(x)-g C f(g(x)-f(x)dx D-|(f(x)-g(x)dx|(1)A：(2)D：(3)D；(4)C：(5)Ao(1)A*=-J/x B (0,1)x2 0(2)F(x)=J；edf,Fr(x)=_ (3)2y=sinxx(4)4-x2dx=_ P=_ J(a0 p0)-X(f ACcos idt cosxcosrdr *2(2)F(x)=-J ddf、:.F(x)=(-Rdf J=-/(.，)=20(2)S=2sinxdr=-2cosx|：=-2(cos-cos0)=4(3)(4)y=-J4-.X24_&=22 (5)pWl(1)(2)Vx(l+4x)dx(3)k(5)jxsin2xdx(1)2_(2-2(2X_A)_2x)|：=4 3 I 7(2)7(1+Vx)rfr=(jx2+-：)lo=f.v2 71-x2 dx Jo x=sint(0 t 0(3)1-H In =f(1+lnx)rf(l+Inx)=-(l+ln.x)1 =i(x|r-sin4z|T)=-(5)P.rsin2xdv=Jo xcos2.r|j+lfcos2.vdx+lsin2.r 71 yr=1y=.v,y=2 y=x,xy=lx=l.y=l iiJB1s”1 A limH0；BSC、lim=0 Dlimo W n JTT8 JTTOC 2n=!0 8 An=l n=!c3a”r=l BW=I D/!=1=1 0 As n=l c(d”+c)H=1 BDS”+e）rr=l 41211=1 An=l n=Ic5?”19=1 1 5Rx）=X%x”a”=B.a”Jl=D/1=1 Ax n=O(/(0)nl c)(0)n 1 Dx n ID 2A 3.B 4.A 5.C 1|q|_=0 3f+1）!limlim1=lim3（）fl=lim=-1 a”Afg 3 円f1 n e（1n 1 1（3）=盼/=1.2 y/n n w+1lima,=lim-=0.E EH 仁 2.4%（l”=lim=limlR=l,an（2）（X-1）2=y,Z?45 4（+1）4 ir=l n（n1）3a”_ ir=0=0 4x）=exx 59 J（.v-3）2 n=0 n=0 1.1 2.3.1.8 A=l 35!n=l n n 1）n2 n=2S 2 4 Ji1A.vy*-(j)3=sin(A：+y)Bxy-5y-j5+6y=x3:Cy-6y=4x+2 D(y*)2+x=0 2Ax:xv?+6y=A*3 B5yr=xve Cy xy+y D=sin(.r+y)3y=0A%y=cex+c2eJ By=Cje+c2e21 C x y=,sinx+c2 cosx Dy=c,sin 2x+c2 cos2x 40X 1 1 A%=+c：Bs y=xc y%C.y=x：DInj=Inx 52y+y=e-xcos.vyxcosx Ay=xex(c*!cosx+c2 sin)By=xex(Cj cosx+c2 sin)C y=cex cosx Dy=eK(c cosx+c2 sin)1A2C 3C 4B 5D 6、V*+p（x）y=q（x）7y-6”+13y=0 X 9 10yy=q”+c*2 o 1y=e*px(.v)ePt)dtd.v+c)232,3 3F=y（0）=0v=sin.vy（7T）=04y=x3+c,x+c.5y=/+cl 0,a/1)y=logflx(a0,a1)sinx,cosx,1anx,cotx arcsinx arccosx arctanx 4.y=ea+x)yu=v2,v=arctanw,w=l+xa5./(-)=x+Vl+x2(x0),/(x)=_ o X f=X=1 X t 1+J1+x2/(x)=-X 2/(x)=+ln(x-2)gx-20Kb(x-2)/0,x2fix3；5 x0,(2,3)Y(3,5 3/(x)0J,/(lux)o/(lux)0lnx030,3 x-3 x3：(I)eych=e2xdx,ey=-e2xc 2(0)=0f=ey=e2x+2 2 2 2 J f 1 y=ex(j sin xe z rfr+c)=(-cosx+c)0c=-1l(-cosx-I)X sinx.y=-(-cosx+c)X y(;r)=0c=-ly=-(-cosx-1)X 2y+5yf+6y=3e2x+52+6=0A=-3,2,=-2.y=ce-3x+c2e2xy*=Aelx20Je2x=3dA=c“2T-4”+4v=e2x sin 5xX2-4A+4=0=2.=2,(Q+c2.v)e-2x 5y=e2x(/lcos5.r+Bsin5x),A=0 B=25 y=e2xsin5x+(c,+c2.r)e2x qg