(a) 画出一对适当函数的图像草图，说明方程 $4-x^2=\sec\tfrac12x$ 在区间 $0\leq x<\pi$ 内恰有一个根。

(b) 通过计算验证此根位于 $1$ 与 $2$ 之间。

(c) 利用迭代公式 $x_{n+1}=\sqrt{4-\sec\tfrac12x_n}$ 求该根，精确到小数点后 2 位。写出每次迭代的结果，精确到小数点后 4 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。

画出严格递减的抛物线右半支与严格递增的正割曲线，并标出唯一交点。两曲线在区间两端的相对位置相反。

Sketch $y=4-x^2$, passing through $(0,4)$ and $(2,0)$ and decreasing on $0<x<\pi$.

On the same axes sketch $y=\sec(x/2)$, starting at $(0,1)$, increasing and tending to $+\infty$ as $x\to\pi^-$.

Mark the intersection of the two curves.

At $x=0$, the parabola is above the secant curve. Near $x=\pi$, it is below it. One curve decreases strictly and the other increases strictly, so they cross exactly once. Thus the equation has $\boxed{\text{one root in }[0,\pi)}$.

构造两边之差，在 $1,2$ 处计算异号值；计算器用弧度。

Let $f(x)=4-x^2-\sec(x/2)$. Then

Since $f$ is continuous on $[1,2]$, a root lies between $1$ and $2$. By part (a), this is the unique root.

按给定公式迭代，内部保留精度，逐项显示四位小数并等待结果稳定。

Starting with $x_0=1.5$, use

in radians.

Hence the root is $\boxed{1.60}$ to 2 decimal places.