10 月 2022 - 第7页共7页

偏导数 $\frac{\partial z}{\partial y}$（B012）

问题

已知函数 $z$ $=$ $f(x, y)$ 在 $(x, y)$ 的某邻域内有定义，且以下选项中的极限均存在，则 $\frac{\partial z}{\partial x}$ $=$ $?$

选项

[A]. $\frac{\partial z}{\partial y}$ $=$ $\lim_{\Delta \rightarrow y}$ $\frac{f(x, y + \Delta y) – f(x, y)}{\Delta y}$

[B]. $\frac{\partial z}{\partial y}$ $=$ $\lim_{\Delta \rightarrow y}$ $\frac{f(x + \Delta + x, y) – f(x, y)}{\Delta x}$

[C]. $\frac{\partial z}{\partial y}$ $=$ $\lim_{\Delta \rightarrow y}$ $\frac{f(x, y + \Delta y) – f(x, y)}{\Delta x}$

[D]. $\frac{\partial z}{\partial y}$ $=$ $\lim_{\Delta \rightarrow y}$ $\frac{f(x, y + \Delta y) + f(x, y)}{\Delta y}$

答案

$\frac{\partial \textcolor{red}{z}}{\partial \textcolor{orange}{y}}$ $=$ $\lim_{\textcolor{yellow}{\Delta} \rightarrow \textcolor{orange}{y}}$ $\frac{f(\textcolor{cyan}{x}, \textcolor{orange}{y} + \textcolor{yellow}{\Delta} \textcolor{orange}{y}) – f(\textcolor{cyan}{x}, \textcolor{orange}{y})}{\textcolor{yellow}{\Delta} \textcolor{orange}{y}}$

偏导数 $\frac{\partial z}{\partial x}$（B012）

问题

已知函数 $z$ $=$ $f(x, y)$ 在 $(x, y)$ 的某邻域内有定义，且以下选项中的极限均存在，则 $\frac{\partial z}{\partial x}$ $=$ $?$

选项

[A]. $\frac{\partial z}{\partial x}$ $=$ $\lim_{\Delta \rightarrow x}$ $\frac{f(x + \Delta x, y) + f(x, y)}{\Delta x}$

[B]. $\frac{\partial z}{\partial x}$ $=$ $\lim_{\Delta \rightarrow x}$ $\frac{f(x + \Delta x, y) – f(x, y)}{f(x, y)}$

[C]. $\frac{\partial z}{\partial x}$ $=$ $\lim_{\Delta \rightarrow x}$ $\frac{f(x + \Delta x, y) – f(x, y)}{x}$

[D]. $\frac{\partial z}{\partial x}$ $=$ $\lim_{\Delta \rightarrow x}$ $\frac{f(x, y + \Delta y) – f(x, y)}{\Delta x}$

答案

$\frac{\partial \textcolor{red}{z}}{\partial \textcolor{orange}{x}}$ $=$ $\lim_{\textcolor{yellow}{\Delta} \rightarrow \textcolor{orange}{x}}$ $\frac{f(\textcolor{orange}{x} + \textcolor{yellow}{\Delta} \textcolor{orange}{x}, \textcolor{cyan}{y}) \textcolor{yellow}{-} f(\textcolor{orange}{x}, \textcolor{cyan}{y})}{\textcolor{yellow}{\Delta} \textcolor{orange}{x}}$

空间曲线在 $zOx$ 平面上的投影曲线的方程（B011）

问题

已知空间曲线 $L$ 的一般方程为 $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$, 则该曲线在空间直角坐标系的 $zOx$ 平面上的投影曲线的方程该如何表示？

选项

[A]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $z$ $\Rightarrow$ $\left\{\begin{array}{l} T(x, y)=0 \\ z=0 \end{array}\right.$

[B]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $y$ $\Rightarrow$ $\left\{\begin{array}{l} T(x, z)=0 \\ y=0 \end{array}\right.$

[C]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $x$ $\Rightarrow$ $\left\{\begin{array}{l} T(y, z)=0 \\ x=0 \end{array}\right.$

[D]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $y$ $\Rightarrow$ $\left\{\begin{array}{l} T(x, z)=y \\ y=0 \end{array}\right.$

答案

$\left\{\begin{array}{l} F(\textcolor{orange}{x}, \textcolor{orange}{y}, \textcolor{cyan}{z})=0 \\ G(\textcolor{orange}{x}, \textcolor{orange}{y}, \textcolor{cyan}{z})=0 \end{array}\right.$ $\Rightarrow$ 消去 $\textcolor{cyan}{y}$ $\Rightarrow$ $\left\{\begin{array}{l} T(\textcolor{orange}{x}, \textcolor{orange}{z})=0 \\ \textcolor{cyan}{y}=\textcolor{red}{0} \end{array}\right.$

空间曲线在 $yOz$ 平面上的投影曲线的方程（B011）

问题

已知空间曲线 $L$ 的一般方程为 $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$, 则该曲线在空间直角坐标系的 $yOz$ 平面上的投影曲线的方程该如何表示？

选项

[A]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $x$ $\Rightarrow$ $\left\{\begin{array}{l} R(y, z)=0 \\ x=1 \end{array}\right.$

[B]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $x$ $\Rightarrow$ $\left\{\begin{array}{l} R(y, z)=x \\ x=0 \end{array}\right.$

[C]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $y$ $\Rightarrow$ $\left\{\begin{array}{l} R(x, z)=0 \\ x=0 \end{array}\right.$

[D]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $x$ $\Rightarrow$ $\left\{\begin{array}{l} R(y, z)=0 \\ x=0 \end{array}\right.$

答案

$\left\{\begin{array}{l} F(\textcolor{orange}{x}, \textcolor{orange}{y}, \textcolor{cyan}{z})=0 \\ G(\textcolor{orange}{x}, \textcolor{orange}{y}, \textcolor{cyan}{z})=0 \end{array}\right.$ $\Rightarrow$ 消去 $\textcolor{cyan}{x}$ $\Rightarrow$ $\left\{\begin{array}{l} R(\textcolor{orange}{y}, \textcolor{orange}{z})=0 \\ \textcolor{cyan}{x}=\textcolor{red}{0} \end{array}\right.$

空间曲线在 $xOy$ 平面上的投影曲线的方程（B011）

问题

已知空间曲线 $L$ 的一般方程为 $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$, 则该曲线在空间直角坐标系的 $xOy$ 平面上的投影曲线的方程该如何表示？

选项

[A]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $y$ $\Rightarrow$ $\left\{\begin{array}{l} H(x, y)=0 \\ z=0 \end{array}\right.$

[B]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $y$ $\Rightarrow$ $\left\{\begin{array}{l} H(x, z)=0 \\ z=0 \end{array}\right.$

[C]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $z$ $\Rightarrow$ $\left\{\begin{array}{l} H(x, y)=0 \\ z=0 \end{array}\right.$

[D]. $\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.$ $\Rightarrow$ 消去 $z$ $\Rightarrow$ $\left\{\begin{array}{l} H(x, y)=z \\ z=0 \end{array}\right.$

答案

$\left\{\begin{array}{l} F(\textcolor{orange}{x}, \textcolor{orange}{y}, \textcolor{cyan}{z})=0 \\ G(\textcolor{orange}{x}, \textcolor{orange}{y}, \textcolor{cyan}{z})=0 \end{array}\right.$ $\Rightarrow$ 消去 $\textcolor{cyan}{z}$ $\Rightarrow$ $\left\{\begin{array}{l} H(\textcolor{orange}{x}, \textcolor{orange}{y})=0 \\ \textcolor{cyan}{z}=\textcolor{red}{0} \end{array}\right.$