Locating Roots | 根的定位

📚 Locating Roots | 根的定位

In A-Level Mathematics, locating roots means finding the values of x for which a function f(x) equals zero. These values are called roots, solutions, or x-intercepts. This topic is central to Edexcel Pure Mathematics and forms the foundation for numerical methods such as iteration and the Newton-Raphson method.

在 A-Level 数学中，根的定位是指求出使函数 f(x) 等于零的 x 值。这些值称为根、解或 x 轴截距。该主题是 Edexcel 纯数学的核心内容，也是迭代法、牛顿-拉弗森法等数值方法的基础。

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1. What is a root? | 什么是根？

For a function y = f(x), a root is any value x = α such that f(α) = 0.

对于函数 y = f(x)，根是满足 f(α) = 0 的任意值 x = α。

Graphically, a root is a point where the curve y = f(x) crosses or touches the x-axis. The root is the x-coordinate of that point, not the point itself.

从图像上看，根是曲线 y = f(x) 与 x 轴相交或相切的点。根是该点的 x 坐标，而不是点本身。

• Example: f(x) = x² – 9 has roots x = -3 and x = 3.
• 例如：f(x) = x² – 9 的根为 x = -3 和 x = 3。

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2. The sign-change rule | 变号法则

If f(x) is continuous on the interval [a, b] and f(a) × f(b) < 0, then f(x) has at least one root in the interval (a, b).

如果 f(x) 在区间 [a, b] 上连续，且 f(a) × f(b) < 0，则 f(x) 在区间 (a, b) 内至少有一个根。

f(a) × f(b) < 0 ⇒ at least one root in (a, b)

f(a) × f(b) < 0 ⇒ 在 (a, b) 内至少有一个根

The product is negative only when f(a) and f(b) have opposite signs. This means the curve must cross the x-axis somewhere between a and b.

只有当 f(a) 和 f(b) 异号时，乘积才为负。这意味着曲线必定在 a 与 b 之间的某处穿过 x 轴。

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3. Conditions for the rule to work | 法则适用条件

The sign-change rule requires two things: the function must be continuous on [a, b], and we must be able to evaluate f(a) and f(b).

变号法则需要两个条件：函数在 [a, b] 上必须连续，并且我们能够计算 f(a) 和 f(b)。

Continuity means there are no breaks, jumps, vertical asymptotes, or holes in the graph on the interval. A continuous curve can cross the x-axis without being lifted from the paper.

连续意味着在该区间内图像没有断裂、跳跃、垂直渐近线或空洞。连续曲线可以不离开纸面地穿过 x 轴。

If f(a) × f(b) > 0, there may still be roots, but the sign-change rule cannot detect them because an even number of roots could occur between a and b.

如果 f(a) × f(b) > 0，区间内仍可能存在根，但变号法则无法检测到，因为 a 与 b 之间可能出现偶数个根。

Sign of f(a) × f(b)	Conclusion for continuous f(x)
f(a) × f(b) < 0	At least one root in (a, b)
f(a) × f(b) > 0	No conclusion: zero, two, or more roots may exist
f(a) × f(b) = 0	a or b is a root

For example, f(x) = x² – 4x + 3 has roots at x = 1 and x = 3. On [0, 4], f(0) = 3 and f(4) = 3, so the product is positive, yet two roots exist.

例如，f(x) = x² – 4x + 3 在 x = 1 和 x = 3 处有根。在区间 [0, 4] 上，f(0) = 3 且 f(4) = 3，乘积为正，但依然存在两个根。

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4. Worked example: locating a cubic root | 例题：定位三次方程根

Consider f(x) = x³ – x – 2. We want to show that a root lies between x = 1 and x = 2.

考虑 f(x) = x³ – x – 2。我们要证明在 x = 1 与 x = 2 之间存在一个根。

Evaluate: f(1) = 1³ – 1 – 2 = -2 and f(2) = 2³ – 2 – 2 = 4.

计算：f(1) = 1³ – 1 – 2 = -2，f(2) = 2³ – 2 – 2 = 4。

Since f(1) < 0 and f(2) > 0, the product f(1) × f(2) = -8 < 0. Therefore, there is at least one root in (1, 2).

因为 f(1) < 0 且 f(2) > 0，所以 f(1) × f(2) = -8 < 0。因此，在 (1, 2) 内至少存在一个根。

This result only tells us that at least one root exists. It does not give the exact value or the number of roots in the interval.

这个结果只告诉我们至少存在一个根，并不给出精确值或区间内根的个数。

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5. Narrowing the interval by decimal search | 用小数搜索缩小区间

To find a root to a required accuracy, we can test values at smaller steps and look for a sign change.

为了以所需精度求根，我们可以用更小的步长测试函数值并寻找变号。

For f(x) = x³ – x – 2, try x = 1.5: f(1.5) = 3.375 – 3.5 = -0.125. Try x = 1.6: f(1.6) = 4.096 – 3.6 = 0.496.

对于 f(x) = x³ – x – 2，尝试 x = 1.5：f(1.5) = 3.375 – 3.5 = -0.125。尝试 x = 1.6：f(1.6) = 4.096 – 3.6 = 0.496。

The sign changes between 1.5 and 1.6, so the root lies in (1.5, 1.6). Continuing with steps of 0.01 gives a root near 1.52.

符号在 1.5 与 1.6 之间改变，因此根位于 (1.5, 1.6)。继续以 0.01 为步长可得到根约为 1.52。

x	f(x) = x³ – x – 2
1.50	-0.125
1.51	-0.066049
1.52	-0.008192
1.53	0.051377

The root lies between 1.52 and 1.53 because f(1.52) < 0 and f(1.53) > 0. This gives the root to 1 decimal place as 1.5.

根位于

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