1.计算下列极限 (每小题7分,共28分)
(1).$displaystyle limlimits_{x o 0 } frac{sqrt{cos x}-sqrt[3]{cos x}}{sin^{2}x}$.
(2)$displaystyle limlimits_{n o infty} left( frac{1^{p}+2^{p}+cdots +n^{p}}{n^{p}}-frac{n}{p+1}
ight)$,$(pin N,pge 1)$.
(3)$displaystyle limlimits_{x o +infty}frac{displaystyle x^{2ln x}-x}{(ln x)^{x}+x}$.
(4)$displaystyle limlimits_{n o infty}sqrt[n]{1+a^{n}+sin^{2}n},a>0$.
2.计算积分(每小题8分,共40分)
(1).$displaystyle intfrac{sin x cos^{3}x}{1+cos^{2}x}dx$;
(2).$displaystyle oint_{L}frac{(x-y)dx+(x+4y)dy}{x^{2}+4y^{2}}$,其中$L$为单位圆$x^{2}+y^{2}=1$,取逆时针方向;
(3).$displaystyle iint_{Sigma}(2x+z)dydz+zdxdy$,其中$Sigma$是曲面$z=x^{2}+y^{2}(0le zle 1)$取上侧;
(4).$displaystyle g(alpha)=int_{1}^{+infty}frac{arctan alpha x}{x^{2}sqrt{x^{2}-1}}dx$;
(5).设$displaystyle f(x)=int_{1}^{x}frac{sin t}{t}dt$,求$displaystyle int_{0}^{1}xf(x)dx$.
3.(本题12分) $f(x)$在$[0,+infty)$连续,且$displaystyle limlimits_{x o +infty}left(f(x)+sin x ight)=0$.证明:$f(x)$在$[0,+infty)$上一致连续.
4.(本题10分) 令$u=f(z)$,其中$z=z(x,y)$是由方程$displaystyle z=x+yvarphi(z)$确定的隐函数,且$f(z)$和$varphi (z)$是任意阶可微函数.证明:
$$frac{partial ^{n}u}{partial y^{n}}=frac{partial ^{n-1}}{partial x^{n-1}}left[ left(varphi(z) ight)^{n}frac{partial u}{partial x} ight]$$
5.(本题15分) 证明:如果函数$f(x)$在$(0,+infty)$内可微,且$displaystyle limlimits_{x o +infty}f'(x)=0$.则$displaystyle limlimits_{x o +infty}frac{f(x)}{x}=0$.
6.(本题10分) 设$f(x)$在$[a,b]$内可导且$f(a)=0$.证明:$$M^{2}le (b-a)int_{a}^{b}left[f'(x) ight]^{2}dx$$其中,$M=suplimits_{ale xle b}left{ left|f(x) ight| ight}$.
7.(本题20分)
设$displaystyle f_{n}(x)=n^{alpha}xe^{-nx},nin N$.当参数$alpha $为何值时
(1).函数列$displaystyle {f_{n}(x)}$在$[0,1]$上收敛;
(2).函数列$displaystyle {f_{n}(x)}$在$[0,1]$上一致收敛;
(3).$displaystyle int_{0}^{1}limlimits_{n o infty}f_{n}(x)dx=limlimits_{n oinfty}int_{0}^{1}f_{n}(x)dx$.
8.(本题15分) 证明:$$displaystyle iiintlimits_{Omega}frac{dxdydz}{r}=frac{1}{2}iintlimits_{partial Omega}cos left( overrightarrow{r}, overrightarrow{n} ight)dS$$
其中$Omega$为$R^{3}$中的单连通区域,$partial Omega$为其光滑边界曲面,$overrightarrow{n}$为$partial Omega$在点$(x,y,z)$的单位外发矢量,$r=sqrt{(xi -x)^{2}+(eta -y)^{2}+(zeta -z)^{2}},overrightarrow{r}=(x-xi)overrightarrow{i}+(y-eta)overrightarrow{j}+(z-zeta)overrightarrow{k}$为连接空间中点$(xi,eta,zeta)$到$(x,y,z)$的矢量.