📝 题目
4．求函数 $z=\mathrm{e}^{x y}$ 当 $x=1, y=1, \Delta x=0.15, \Delta y=0.1$ 时的全微分．
💡 答案与解析
[AI解答]
[AI解答] 函数为 $$ z = e^{xy} $$ 先求偏导数： $$ \frac{\partial z}{\partial x} = y e^{xy}, \quad \frac{\partial z}{\partial y} = x e^{xy} $$ 全微分公式为 $$ \mathrm{d}z = \frac{\partial z}{\partial x} \mathrm{d}x + \frac{\partial z}{\partial y} \mathrm{d}y $$ 代入给定点 $x=1, y=1$ 得 $$ \left.\frac{\partial z}{\partial x}\right|_{(1,1)} = 1 \cdot e^{1} = e,\quad \left.\frac{\partial z}{\partial y}\right|_{(1,1)} = 1 \cdot e^{1} = e $$ 因此全微分为 $$ \mathrm{d}z = e \cdot \Delta x + e \cdot \Delta y $$ 代入 $\Delta x = 0.15,\ \Delta y = 0.1$： $$ \mathrm{d}z = e (0.15 + 0.1) = 0.25 e $$ 若取 $e \approx 2.71828$，则数值近似为 $$ \mathrm{d}z \approx 0.25 \times 2.71828 = 0.67957 $$
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