从第一性原理求 $e^x\tan x$ 的麦克劳林级数，展开至包含 $x^2$ 项为止。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

从原函数直接求导。计算零点的函数值、一阶导数和二阶导数，再代入 Maclaurin 公式。

Let $f(x)=e^x\tan x$. The product and chain rules give

Here the second derivative comes from $f''=e^x(\tan x+\sec^2x)+e^x(\sec^2x+2\sec^2x\tan x)$. Thus $f(0)=0$, $f'(0)=1$, and $f''(0)=2$. The Maclaurin formula is

Substitution gives $0+x+\frac{x^2}{2}(2)+O(x^3)$, so $\boxed{e^x\tan x=x+x^2+O(x^3)}$.