设 $\mathrm{f}(x)=\dfrac{2\mathrm{e}^{2x}}{\mathrm{e}^{2x}-3\mathrm{e}^x+2}$。

(a) 求 $\mathrm{f}'(x)$，并由此求方程为 $y=\mathrm{f}(x)$ 的曲线的驻点的精确坐标。

(b) 使用代换 $u=\mathrm{e}^x$ 和部分分式，求 $\displaystyle\int_{\ln3}^{\ln5}\mathrm{f}(x)\,\mathrm{d}x$ 的精确值。

答案以 $\ln a$ 的形式表示，其中 $a$ 为最简形式的有理数。

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。

用商法则求导，分子约去相消项后只需求一个指数值。原分母为零的两点要排除。

The domain excludes $e^x=1,2$, so $x\ne0,\ln2$.

Write $D=e^{2x}-3e^x+2$, so $D'=2e^{2x}-3e^x$. The quotient rule gives

The numerator simplifies as follows:

Hence

Since $e^{2x}>0$, the stationary point requires $4-3e^x=0$, giving $x=\ln(4/3)$.

At this value,

Thus the exact stationary point is

也可先把函数按 $e^x$ 做部分分式，再分别求导，最后通分求驻点。

Let $u=e^x$. Polynomial division and partial fractions give

Indeed, the remaining numerator $6u-4$ equals $8(u-1)-2(u-2)$.

Thus

Differentiating,

At a stationary point, divide by $2e^x>0$:

Cross-multiplication gives

Since $u>0$, $u=4/3$. Substituting in $f$ gives $y=-16$, so the stationary point is

按题意换元时，微分带来一个 $u$ 的分母，积分中的分子由 $2u^2$ 变成 $2u$。然后拆分两项简单对数。

Let $u=e^x$, so $\mathrm{d}u=u\,\mathrm{d}x$. The limits become $u=3$ and $u=5$.

Hence

Write

Then $2u=A(u-2)+B(u-1)$. Putting $u=1$ gives $A=-2$, and putting $u=2$ gives $B=4$.

On $3\leqslant u\leqslant5$, both $u-1$ and $u-2$ are positive. Therefore