(a) 通过画出一对合适的图像的草图，证明方程 $2+\mathrm{e}^{-0.2x}=\ln(1+x)$ 只有一个根。

(b) 通过计算证明此根位于 $7$ 与 $9$ 之间。

(c) 使用迭代公式

求此根，结果保留 $2$ 位小数。每次迭代的结果保留 $4$ 位小数。

[$\exp(x)$ 是 $\mathrm{e}^x$ 的另一种记法。]

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。

对数的定义域为 $x>-1$。在这整个定义域，一条曲线递增，另一条递减，且两端的上下关系交换。

Sketch $y=2+e^{-0.2x}$ and $y=\ln(1+x)$ for $x>-1$.

The exponential curve decreases, passes through $(0,3)$ and has horizontal asymptote $y=2$. The logarithmic curve increases, passes through $(0,0)$ and has vertical asymptote $x=-1$.

As $x\to-1^+$, the logarithmic curve is below the exponential curve. As $x\to\infty$, it grows without bound while the exponential curve approaches $2$.

Continuity gives an intersection, and the opposite strict monotonicity ensures that there is exactly one. Therefore the equation has $\boxed{\text{exactly one root}}$.

构造两边之差并检查变号。

Let $f(x)=2+e^{-0.2x}-\ln(1+x)$. Then

Since $f$ is continuous on $[7,9]$, the sign change proves that the root lies in $\boxed{7<x<9}$.

用区间内初值迭代，输出四位小数，内部保留完整精度。

Starting with $x_0=8$, use

This gives

Hence the root is $\boxed{x=8.03}$ to 2 decimal places.