Tangent and Normal Equations | 切线与法线方程

📚 Tangent and Normal Equations | 切线与法线方程

In IB Mathematics, especially in the topic of calculus, you will frequently be asked to find the equations of tangent and normal lines to a given curve. The tangent line touches the curve at a single point with a slope equal to the derivative, while the normal line is perpendicular to the tangent. Understanding these concepts and the steps to derive their equations is essential for success in both Analysis & Approaches and Applications & Interpretation courses.

在IB数学中，尤其是微积分部分，你会经常遇到求曲线切线和法线方程的问题。切线在一点处与曲线相切，其斜率等于导数；法线则垂直于切线。理解这些概念并掌握推导方程的方法，对于在分析与方法、应用与解释两门课程中取得成功至关重要。

1. Understanding Tangents | 理解切线

A tangent line to a curve at a point P is a straight line that touches the curve at P without crossing it locally. It represents the instantaneous direction of the curve at that point.

曲线在点P处的切线是一条在P点与曲线接触（局部不穿过）的直线。它代表了曲线在该点的瞬时方向。

For a function y = f(x), the tangent at x = a is the line that passes through (a, f(a)) with slope equal to the derivative f'(a).

对于函数 y = f(x)，在 x = a 处的切线是经过点 (a, f(a)) 且斜率为导数 f'(a) 的直线。

---

2. Understanding Normals | 理解法线

The normal line to a curve at a point is the line perpendicular to the tangent at that point. It shares the same point of contact with the curve.

曲线在某点的法线是垂直于该点切线的直线，它与曲线经过同一个切点。

If the tangent has slope mₜ = f'(a) (and f'(a) ≠ 0), then the normal slope is the negative reciprocal: mₙ = −1 / f'(a). When f'(a) = 0, the tangent is horizontal (y = f(a)) and the normal is vertical, given by x = a.

如果切线斜率为 mₜ = f'(a)（且 f'(a) ≠ 0），则法线斜率为负倒数：mₙ = −1 / f'(a)。当 f'(a) = 0 时，切线为水平线 y = f(a)，法线为竖直线 x = a。

---

3. The Derivative as Slope | 导数即斜率

The derivative f'(x) represents the rate of change of the function and geometrically gives the slope of the tangent line at any point x. Therefore, confident differentiation is the foundation for finding tangents and normals.

导数 f'(x) 表示函数的变化率，在几何上给出了任意点 x 处切线的斜率。因此，熟练求导是求切线法线的基础。

You must be comfortable using power rule, product rule, quotient rule, and chain rule, as well as implicit and parametric differentiation when required.

你必须熟练掌握幂法则、积法则、商法则和链式法则，并在需要时能运用隐函数求导和参数方程求导。

---

4. Equation of a Tangent Line | 切线方程

Given y = f(x) and a point (x₁, y₁) on the curve with y₁ = f(x₁), the tangent equation is written in point‑slope form:

给定 y = f(x) 及曲线上一点 (x₁, y₁) 满足 y₁ = f(x₁)，切线方程用点斜式表示为：

y − y₁ = f'(x₁)(x − x₁)

The slope of the tangent is exactly the value of the derivative at x₁.

切线的斜率正好是导数在 x₁ 处的值。

---

5. Equation of a Normal Line | 法线方程

Since the normal is perpendicular to the tangent, its slope is the negative reciprocal of the tangent slope, provided f'(x₁) ≠ 0:

因为法线与切线垂直，其斜率是切线斜率的负倒数，前提 f'(x₁) ≠ 0：

y − y₁ = −(1 / f'(x₁)) (x − x₁)

If f'(x₁) = 0, the tangent is y = y₁ and the normal is the vertical line x = x₁.

若 f'(x₁) = 0，切线为 y = y₁，法线为竖直线 x = x₁。

---

6. Step-by-Step Procedure | 分步流程

Step 1: Differentiate the function to obtain f'(x).

步骤一：对函数求导，得到 f'(x)。

Step 2: Substitute the given x‑coordinate into f'(x) to find the tangent slope mₜ = f'(x₁).

步骤二：将给定的 x 坐标代入 f'(x) 求出切线斜率 mₜ = f'(x₁)。

Step 3: Calculate the y‑coordinate if not provided, using y₁ = f(x₁).

步骤三：如果未提供 y 坐标，用 y₁ = f(x₁) 计算。

Step 4: Write the tangent equation using point‑slope form: y − y₁ = mₜ (x − x₁). Simplify if required.

步骤四：用点斜式写出切线方程：y − y₁ = mₜ (x − x₁)。如有需要可化简。

Step 5: For the normal, compute slope mₙ = −1 / mₜ (if mₜ ≠ 0) and then write y − y₁ = mₙ (x − x₁).

步骤五：对于法线，计算斜率 mₙ = −1 / mₜ（若 m

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）