一、极限问题 (每小题8分,共32分)
1.设集合$A
ot =varnothing ,alpha =sup A,alpha
ot in A.$证明:$A$中存在严格单调递增数列${x_{n}}$,满足$limlimits_{n o infty}x_{n}=alpha$.
2.设$x_{0}=a,x_{1}=b(0<a<b)$,且$displaystyle x_{n+1}=sqrt{x_{n}x_{n-1}},(nge1)$.证明:${x_{n}}$收敛,并求$limlimits_{n o infty}x_{n}$.
3.求$displaystyle limlimits_{x o 0}frac{e^{x^{2}}-xsin x-1}{x^{4}}$.
4.求$displaystyle limlimits_{x o 0}frac{sqrt{cos x}-sqrt[3]{cos x}}{ln (x^{2}+1)}$.
二、计算积分(每小题8分,共32分)
1.$displaystyle int_{0}^{1}frac{x^{2001}-x^{1005}}{ln x}dx$;
2.设$displaystyle f(x)$在$[0,1]$可积,且满足$displaystyle x^{2}(ln x)^{2}-f(x)=int_{0}^{1}f(x)dx$,求$displaystyle int_{0}^{1}f(x)dx$的值;
3.计算$displaystyle I=int_{L}left(x^{2}+2y+z ight)ds$,其中$L$是圆周$left(x-2 ight)^{2}+y^{2}=r^{2}(r>0)$取逆时针方向;
4.计算$displaystyle iintlimits_{S}(x+2y)dydz+(y+z)dzdx+(z+2)dxdy $,其中$S$是椭球面$displaystyle frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}+frac{z^{2}}{c^{2}}=1$的上半部分,其方向为下侧;
5.设$f(x)$在$(-infty,+infty)$内有连续导函数,计算积分$displaystyle intlimits_{L}frac{1+y^{2}f(xy)}{y}dx+frac{x}{y^{2}}[y^{2}f(xy)-1]dy$,其中$L$为上半平面$(yge 0)$内以$left(2,3 ight)$为起点到$ left(3,2 ight)$为终点的有向分段光滑曲线.
三、(15分) 设正项级数$displaystyle sumlimits_{n=1}^{infty}a_{n}$发散,且$displaystyle s_{n}=sumlimits_{k=1}^{n}a_{k}$,讨论$displaystylesumlimits_{n=1}^{infty}frac{a_{n}}{s_{n}^{alpha}}$的敛散性,其中$alpha>0$.
四、(15分) 讨论函数egin{equation*}displaystyle f(x,y)=egin{cases}displaystyle (x+y)^{2}sin frac{1}{x^{2}+y^{2}}&,(x,y) ot =(0,0)\0&,(x,y=(0,0)end{cases}end{equation*}的偏导数$f_{x},f_{y}$在原点的连续性和$f$在原点的可微性.
五、(15分) 设$f(x)$在$(0,2)$上二阶可导,$f''(1)>0$.证明:存在$x_{1},x_{2}in (0,2)$,使得$displaystyle f'(1)=frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$.
六、(12分) (15分) 设连续$displaystyle f(x):R o R$在所有无理点处取有理值,且$f(0)=1$,求$f(x)$.
七、(每小题7分,共21分) 设$displaystyle f(x)=int_{1}^{+infty}frac{sin xt}{t(1+t^{2})}dt,xin (-infty,+infty) $.证明:
1.证明积分$displaystyle int_{1}^{+infty}frac{sin xt}{t(1+t^{2})}dt$关于$x$在$(-infty,+infty)$一致收敛;
2.证明$displaystyle limlimits_{x o +infty}f(x)=0$;
3.$f(x)$在$(-infty,+infty)$上一致连续.