IAL 2026 Jan FP1 Q1 | 國際數學站

Question

[!problem]

$f(x) = 3x - \dfrac{5}{x}$

The equation $f(x) = 0$ has a root $\alpha$ in the interval $[3, 3.8]$

(a) Starting with the end points of this interval, use interval bisection to determine $\alpha$ to one decimal place.

(4)

(b) (i) Determine $f'(x)$ .

(ii) Using your answer to part (a) as a first approximation to $\alpha$ , apply the Newton-Raphson process once to calculate a further approximation for $\alpha$ , giving your answer to 4 decimal places.

(2)

中文翻译

$f(x) = 3x - \dfrac{5}{x}$

方程 $f(x) = 0$ 在区间 $[3, 3.8]$ 内有根 $\alpha$

(a) 从该区间的端点开始，使用区间二分法将 $\alpha$ 确定到 1 位小数。

(4)

(b) (i) 求 $f'(x)$ 。

(ii) 使用 (a) 的答案作为 $\alpha$ 的初始近似值，应用一次牛顿-拉夫森过程计算 $\alpha$ 的进一步近似值，答案精确到 4 位小数。

(2)