单选题 （共 6 题 ），每题只有一个选项正确
$\int_0^1 \frac{\arcsin \sqrt{x}}{\sqrt{x(1-x)}} d x=$
$\text{A.}$ $\frac{\pi^2}{4}$.
$\text{B.}$ $\frac{\pi^2}{8}$.
$\text{C.}$ $\frac{\pi}{4}$.
$\text{D.}$ $\frac{\pi}{8}$.
设 $f(x, y)$ 具有一阶偏导数, 且在任意的 $(x, y)$ 都有 $\frac{\partial f(x, y)}{\partial x}>0, \frac{\partial f(x, y)}{\partial y} < 0$,则
$\text{A.}$ $f(0,0)>f(1,1)$.
$\text{B.}$ $f(0,0) < f(1,1)$.
$\text{C.}$ $f(0,1)>f(1,0)$.
$\text{D.}$ $f(0,1) < f(1,0)$.
二元函数 $z=x y(3-x-y)$ 的极值点是
$\text{A.}$ $(0,0)$.
$\text{B.}$ $(0,3)$.
$\text{C.}$ $(3,0)$.
$\text{D.}$ $(1,1)$.
设 $z=z(x, y)$ 由 $\left\{\begin{array}{l}x=u e ^v, \\ y=u v,(u>0, v>1) \\ z=v\end{array}\right.$ 所确定, 则 $\frac{\partial^2 z}{\partial x \partial y}=$
$\text{A.}$ $\frac{x y}{z(1-z)^3}$.
$\text{B.}$ $\frac{x y}{z(z-1)^3}$.
$\text{C.}$ $\frac{z}{x y(1-z)^3}$.
$\text{D.}$ $\frac{z}{x y(z-1)^3}$.
设 $f^{\prime}\left(x_0\right)$ 存在, 则 $\lim _{\Delta x \rightarrow 0} \frac{f\left(x_0-\Delta x\right)-f\left(x_0\right)}{\Delta x}=$.
$\text{A.}$ $f^{\prime}\left(x_0\right)$
$\text{B.}$ $-f^{\prime}\left(x_0\right)$
$\text{C.}$ $2 f^{\prime}\left(x_0\right)$
$\text{D.}$ 不存在
(1) 设 $f(x)$ 满足 $\lim _{x \rightarrow 0} \frac{\sqrt{1+f(x) \sin 2 x}-1}{e^{x^2}-1}=1$, 则( )
$\text{A.}$ $f(0)=0$
$\text{B.}$ $\lim _{x \rightarrow 0} f(x)=0$
$\text{C.}$ $f^{\prime}(0)=1$
$\text{D.}$ $\lim _{x \rightarrow 0} f^{\prime}(x)=1$