07-08期末(1).pdf

120072008?(?5?,?35?)1.limn?pn+3n pn n?;2.limx+x+x1+x;3.limx0?1xcosxex1?;4.rexdx;5.r30 xx+1dx;6.re1xlnxdx;7.rdxex+ex.?1.limn?pn+3n pn n?=limnn+3n(nn)n+3n+n+n=limn4nn+3n+n+n=2.2.limx+x+x1+x=1.3.limx0?1xcosxex1?=limx0ex1xcosxx(ex1)=limx0ex1xcosxx2=limx0excosx+xsinx2x=limx012(ex+2sinx+xcosx)=12.4.rexdx=ret 2tdt=2rtdet=2?tetretdt?=2et(t 1)+C=2ex(x 1)+C.5.r30 xx+1dx=r30(x+1)1x+1d(x+1)=r30hx+1 (x+1)12id(x+1)=h23(x+1)32 2(x+1)12i?30=83.6.re1xlnxdx=re1lnxd?x22?=x22lnx?e1re1x221xdx=e22re1x2dx=e22?e2414?=e2+14.7.rdxex+ex=rdex(ex)2+1=arctanex+C(?arctanex+C)?7?,?35?1.I=limx?sin1x+cos1x?x.?I=limx?sin1x+cos1x?x=limx0(sinx+cosx)1x=elimx0ln(sin x+cos x)x=elimx0cos xsinxsin x+cos x=e.2?I=limx?sin1x+cos1x?x=elimxln(sin1x+cos1x)1x=elimx(cos1xsin1x)(1x2)1x2=e?I=limx?sin1x+cos1x?x2=limx?1+sin2x?x2=elimxx2sin2x=e2.a,b?f(x)=(ax+bcosx,x 0sinxx x,x 0?x=0?.?f(0)=b,f(0+)=limx0+?sinxx x?=1,?f(x)?x=0?f(0)=f(0+).?b=1.?f(x)=(ax+cosx,x 0sinxx x,x 0?f(0)=limx0(ax+cosx)1x=a+limx0cosx1x=a,f+(0)=limx0+sin xxx1x=limx0+?sinxxx2 1?=limx0+sinxxx2 1=limx0+cosx12x 1=1,?f(x)?x=0?f(0)=f+(0).?a=1.?,a=1,b=1.3.?exy=x+y?y=y(x)?(0,1)?.3?,?x=0?,y=1.?exy=x+y?exy(y+xy)=1+y.?x=0?1=1+y(0).?y(0)=0.?x?exy(y+xy)2+exy(2y+xy)=y.?x=0?1+2y(0)=y(0).?y(0)=1.4.y=f(lnx)ef(x)?f(x)?.?y=f(lnx)1x ef(x)+f(lnx)ef(x)f(x)=ef(x)?f(lnx)x+f(lnx)f(x)?.?dy=ef(x)hf(lnx)x+f(lnx)f(x)idx.5.?f(x)?limx0f(x)x=A,(x)=r10f(xt)dt,(x)?(x)?x=0?.?limx0f(x)x=A,?limx0f(x)=0.?f(x)?f(0)=0.?u=xt,t=ux,dt=1xdu.t:0 1,u:0 x.(x)=w10f(xt)dt=1xwx0f(u)du.(0)=r10f(0 t)dt=0.?(0)=limx01xrx0f(u)du 0 x=limx0rx0f(u)dux2=limx0f(x)2x=A2.4?x 6=0?(x)=?1xwx0f(u)du?=1x2wx0f(u)du+f(x)x.limx0(x)=limx0?1x2wx0f(u)du+f(x)x?=limx0rx0f(u)dux2+A=limx0f(x)2x+A=A2.?(x)?x=0?,(x)=(f(x)x1xrx0f(u)du,x 6=0A2,x=0.?6?f(x)?0,1?(0,1)?f(1)=kw1k0 xe1xf(x)dx(k 1)?(0,1),?f()=(1 1)f().?F(x)=xe1xf(x),?F(1)=f(1).?F(1)=f(1)=kw1k0 xe1xf(x)dx=k 1k F(c)=F(c)c (0,1k)(0,1),k 1.?Rolle?,(c,1)(0,1),?F()=0.?f()=(1 1)f().?12?C=aq2+bq+c0,?q=1e(d p),p?,a,b,c0,d,e?d b.?1?2?1?.?(1)?q=1e(d p)?p=d eq.?L=R C=dq eq2 aq2 bq c0=(e+a)q2+(d b)q c0.5?L=2(e+a)q+(d b)=0?q=db2(e+a).?L=(e+a)(d b)24(e+a)2+(d b)d b2(e+a)c0=(d b)24(e+a)c0.?L=2(e+a)0)?y=lnx?(x0,y0)?.(1)?a?(x0,y0).(2)?x?x?Vx.?(1)?y=ax?y=a2x.?y=lnx?y=12x.?a2x=12x.?x=1a2.?y=ax?y=1.?y=lnx?x=e2.?(e2,1),?a=1e.(2)?y=1ex?x=e2=y2,?.?y=lnx?x=e2y,?.?Vx=we20?xe?2dx we21(lnx)2dx#=?e22we21(lnx)2dx?6?t=x,x=t2,dx=2tdt.x:1 e2,t:1 e.we21(lnx)2dx=we1(lnt)2 2tdt=we1(lnt)2d(t2)=(tlnt)2?e1we1t2 2lnt 1tdt=e2we12tlntdt=e2we1lntd(t2)=e2 t2lnt?e1+we1t2dlnt=we1t21tdt=t22?e1=e2 12.?,Vx=?e22e212?=2.