5. Normal to a Curve | 曲线的法线

📚 5. Normal to a Curve | 曲线的法线

In calculus, once you understand how to find the tangent line to a curve, the concept of the normal line follows logically. The normal to a curve at a given point is the line that is perpendicular to the tangent at that point. This article explores the definition, slope relationship, and step-by-step methods for finding equations of normals for explicit, implicit, and parametric curves. Examples and common pitfalls are included to help you master this essential topic in the IB Mathematics syllabus.

在微积分中，当你学会如何求曲线的切线之后，法线的概念便顺理成章地出现。曲线在某一点的法线就是与该点切线垂直的直线。本文探讨法线的定义、斜率关系以及求解显函数、隐函数和参数曲线法线方程的步骤。文中包含例题和常见错误分析，帮助你掌握 IB 数学大纲中这一重要知识点。

1. What is a Normal to a Curve? | 什么是曲线的法线？

er a point P(x₁, y₁) on the curve. The tangent line at P touches the curve and has a slope equal to the derivative f ' ght line that passes through P and is perpendicular to the tangent line. While the tangent represents the instantaneous direction of the curve, the normal cuts through the curve at a right angle.

对于由 y = f(x) 定义的曲线，设 P(x₁, y₁) 为曲线上一点。该点处的切线与曲线相切，其斜率等于导数 f '(x₁)。P 点处的法线是经过 P 且与切线垂直的直线。切线代表曲线在该点的瞬时方向，而法线则以直角穿过曲线。

2. Relationship Between Normal and Tangent | 法线与切线的关系

If the slope of the tangent at P is m_T = f '(x₁) and m_T ≠ 0, then the slope of the normal m_N satisfies m_T × m_N = −1. Hence, m_N = −1 / m_T. For the special cases: when the tangent is horizontal (m_T = 0), the normal is a vertical line (equation x = x₁); when the tangent is vertical (m_T fined), the normal is horizontal (y = y₁). The table below summarises these relationships.

如果 P 点切线的斜率 m_T = f '(x₁) 且 m_T ≠ 0，那么法线的斜率 m_N 满足 m_T × m_N = −1。因此 m_N = −1 / m_T。特殊情况：当切线水平（m_T = 0）时，法线为垂直线（方程为 x = x₁）；当切线垂直（m_T 无定义）时，法线为水平线（y = y₁）。下表概括了这些关系。

3. Steps to Find the Equation of the Normal | 求法线方程的步骤

To determine the normal line for a curve given explicitly as y = f(x), follow these steps:

1. Identify or compute the coordinates of the point P(x₁, y₁) on the curve. If only x₁ is given, substitute into f(x) to obtain y₁.
2. Differentiate the function to find f '(x), then evaluate the tangent slope m_T = f '(x₁).
3. Calculate the normal slope m_N = −1 / m_T (provided m_T ≠ 0). If m_T = 0, the normal is x = x₁.
4. Use the point-slope form to write the normal equation: y − y₁ = m_N(x − x₁). Simplify to slope-intercept or general form as requested.

对于显函数 y = f(x) 给出的曲线，求法线方程可遵循以下步骤：

1. 确定或计算曲线上点 P(x₁, y₁) 的坐标。若只给出 x₁，代入 f(x) 求得 y₁。
2. 求导得到 f '(x)，然后计算切线斜率 m_T = f '(x₁)。
3. 计算法线斜率 m_N = −1 / m_T（假设 m_T ≠ 0）。若 m_T = 0，法线为 x = x₁。
4. 使用点斜式写出法线方程：y − y₁ = m_N(x − x₁)。按要求化简为斜截式或一般式。

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4. Example 1: Polynomial Function | 示例1：多项式函数

= 0, so the point is (2, 0). Differentiate: dy/dx = 2x − 3. At x = 2, the tangent slope m_T = 2×2 − 3 = 1. Therefore, the normal slope m_N −1. Using point-slope form: y − 0 = −1(x − 2) → y = −x + 2. The normal line is y = −x + 2.

求曲线 y = x² − 3x + 2 在 x = 2 处的法线方程。首先计算 y 坐标：y₁ = 2² − 3×2 + 2 = 0，点为 (2,0)。求导得 dy/dx = 2x − 3。在 x = 2 处，切线斜率 m_T = 1。因此法线斜率 m_N = −1。利用点斜式：y − 0 = −1(x − 2) → y = −x + 2。故法线方程为 y = −x + 2。

5. Example 2: Function with a Radical | 示例2：含根式的函数

Determine the normal to y = √x at x = 4. The point is (4, √4) = (4, 2). The derivative is dy/dx = 1/(2√x). Evaluate at x = 4: m_T = 1/(2√4) = 1/(2×2) = 1/4. Then m_N / (1/4) = −4. The normal equation is y − 2 = −4(x − 4) → y − 2 = −4x + 16 → y = −4x + 18.

求 y = √x 在 x = 4 处的法线。点为 (4, 2)。导数 dy/dx = 1/(2√x)。在 x = 4 处，m_T = 1/(2×2) = 1/4。法线斜率 m_N = −4。法线方程为 y − 2 = −4(x − 4) → y = −4x + 18。

6. Example 3: Trigonometric Function | 示例3：三角函数

he point is (π/2, sin(π/2)) = (π/2, 1). Differentiating gives dy/dx = cos x. At x = π/2, m_T tical line through the point, so its equation is simply x = π/2. Note that we cannot use m_N = −1/0; the vertical line is the direct consequence.

求 y = sin x 在 x = π/2 处的法线。点为 (π/2, 1)。求导得 dy/dx = cos x。在 x = π/2 处，m_T = cos(π/2) = 0。切线水平，因此法线为过该点的垂直线，方程即为 x = π/2。注意此处不能使用 −1/0 的斜率公式，垂直线是直接推论。

7. Normal to a Curve Defined Implicitly | 隐函数曲线的法线

se implicit differentiation to find dy/dx. At a point (x₁, y₁), we evaluate dy/dx to get m_T, then m_N = −1 / m_T ² = 25 at (3, 4). Differentiate both sides: 2x + 2y(dy/dx) = 0 → dy/dx = −x/y. At (3,4), m_T = −3/4. Thus m_N uation: y − 4 = (4/3)(x − 3) → y = (4/3)x. (Multiply out: passes through (3,4) and (0,0)).

当曲线由隐式关系 F(x, y) = 0 给出时，我们使用隐函数求导法求 dy/dx。在点 (x₁, y₁) 处代入得到 m_T，然后 m_N = −1 / m_T。例：圆 x² + y² = 25 在点 (3,4)。两边求导：2x + 2y(dy/dx) = 0 → dy/dx = −x/y。在 (3,4) 处 m_T = −3/4，故 m_N = 4/3。法线方程：y − 4 = (4/3)(x − 3) → y = (4/3)x（经化简，该直线通过原点）。

8. Normal to a Parametric Curve | 参数方程曲线的法线

n by (dy/dt) / (dx/dt). At a parameter t = t₀, find the point (x(t₀), y(t₀)) and compute m_T. Then m_N = −1 / m_T (if m_T ≠ 0). Example: x = t², y = 2t at t = 1. Point: (1², 2×1) = (1, 2). dx/dt = 2

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