一条曲线的方程为 $y=\mathrm{e}^{(x^2+4x)}\cos^2x$，其中 $\cos x\ne0$。

(a) 证明这条曲线驻点的 $x$ 坐标满足方程 $\tan x=x+2$。

(b) 通过画出合适的两个函数图像的草图，证明方程 $\tan x=x+2$ 在区间 $-\pi<x<\pi$ 内只有一个根。

(c) 通过计算证明这个根位于 $1.2$ 与 $1.3$ 之间。

(d) 使用迭代公式 $x_{n+1}=\tan^{-1}(x_n+2)$ 计算这个根，答案精确到小数点后 $3$ 位。每次迭代的结果保留小数点后 $5$ 位。

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。

使用乘积法则，分别对指数因子和 $\cos^2x$ 求导。因题设 $\cos x\ne0$，化简驻点条件时可以除以 $\cos x$。

Differentiating gives

Since $e^{x^2+4x}>0$ and $\cos x\ne0$, setting the derivative to zero is equivalent to

Thus the stationary points satisfy $\tan x=x+2$.

也可对 $y$ 取对数。因为 $\cos x$ 可以为负，应使用 $\ln|\cos x|$ 表示平方的对数。

Since $y>0$,

Logarithmic differentiation gives

At a stationary point the left side is zero, giving $\tan x=x+2$.

画正切函数的三段及直线 $y=x+2$。两侧分支与直线不相交，中间分支从直线下方穿至上方，只有一个交点。

Sketch $y=\tan x$ on $-\pi<x<\pi$, with vertical asymptotes $x=\pm\pi/2$, and the line $y=x+2$, which has intercept $2$ and slope $1$.

The left branch of the tangent graph lies above the line, and the right branch lies below it. On the central branch there is one intersection at a positive $x$. To verify the comparison, put $f(x)=\tan x-x-2$. On each branch, $f'(x)=\tan^2x\ge0$, with strict increase over every nontrivial interval. On the left branch its limiting value at $-\pi$ is $\pi-2>0$, and on the right branch its limiting value at $\pi$ is $-\pi-2<0$. On the central branch $f$ increases from $-\infty$ to $+\infty$. Thus the sketch has exactly one intersection in the stated interval.

在根的两侧计算同一个连续函数的值。符号相反即可用连续性确认区间内有根。

With $f(x)=\tan x-x-2$ and angles in radians,

The function is continuous on $[1.2,1.3]$, so the root lies between $1.2$ and $1.3$.

从已知区间端点开始，按反正切公式迭代。计算保留内部精度，表中结果写五位小数，直到足以确定三位小数的根。

Starting with $x_0=1.2$ and using radians, the iterations are

The root is $1.274$ to three decimal places.