📝 题目
3．求函数 $z=\frac{y}{x}$ 当 $x=2, y=1, \Delta x=0.1, \Delta y=-0.2$ 时的全增量和全微分．
💡 答案与解析
[AI解答]
[AI解答]
**第一步：计算全增量** 全增量定义为 $$ \Delta z = f(x+\Delta x, y+\Delta y) - f(x, y) $$ 代入 $f(x,y)=\frac{y}{x}$，已知 $x=2, y=1, \Delta x=0.1, \Delta y=-0.2$，则 $$ x+\Delta x = 2+0.1 = 2.1, \quad y+\Delta y = 1 + (-0.2) = 0.8 $$ 于是 $$ \Delta z = \frac{0.8}{2.1} - \frac{1}{2} $$ 先计算 $\frac{0.8}{2.1} = \frac{8}{21}$，$\frac{1}{2} = \frac{1}{2}$，通分： $$ \frac{8}{21} - \frac{1}{2} = \frac{16}{42} - \frac{21}{42} = -\frac{5}{42} $$ 因此全增量为 $$ \boxed{\Delta z = -\frac{5}{42}} $$
**第二步：计算全微分** 全微分公式为 $$ \mathrm{d}z = \frac{\partial z}{\partial x}\,\mathrm{d}x + \frac{\partial z}{\partial y}\,\mathrm{d}y $$ 先求偏导数： $$ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}\left(\frac{y}{x}\right) = -\frac{y}{x^2}, \quad \frac{\partial z}{\partial y} = \frac{1}{x} $$ 在点 $(x,y)=(2,1)$ 处： $$ \left.\frac{\partial z}{\partial x}\right|_{(2,1)} = -\frac{1}{2^2} = -\frac{1}{4}, \quad \left.\frac{\partial z}{\partial y}\right|_{(2,1)} = \frac{1}{2} $$ 取 $\mathrm{d}x = \Delta x = 0.1$，$\mathrm{d}y = \Delta y = -0.2$，则 $$ \mathrm{d}z = \left(-\frac{1}{4}\right)\times 0.1 + \frac{1}{2}\times (-0.2) = -0.025 - 0.1 = -0.125 $$ 即 $$ \boxed{\mathrm{d}z = -0.125} $$
**难度评级**：★☆☆☆☆ （直接套用全增量与全微分定义，计算简单，无复杂技巧。）