Tangent and Normal Equations | 切线与法线方程

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

In differential calculus, the tangent line to a curve at a given point represents the instantaneous direction of the curve, while the normal line is perpendicular to the tangent at that point. This article covers how to derive both equations, a core topic for IB Mathematics.

在微分学中，曲线在某一点的切线表示曲线在该点的瞬时方向，而法线则是过该点与切线垂直的直线。本文系统讲解如何求切线与法线方程，这是 IB 数学的核心考点。

1. Derivative as Slope | 导数作为斜率

The derivative \( f'(x) \) describes the rate of change of \( f(x) \) at a given value of \( x \). Geometrically, \( f'(a) \) is the slope of the tangent line to the curve \( y = f(x) \) at the point \( (a, f(a)) \).

导数 \( f'(x) \) 表示函数 \( f(x) \) 在给定 \( x \) 值处的变化率。从几何上看，\( f'(a) \) 就是曲线 \( y = f(x) \) 在点 \( (a, f(a)) \) 处切线的斜率。

If the derivative does not exist at a point, the tangent line may still have a limit (vertical tangent) or not exist at all. In IB, you are usually asked to find equations where the derivative is well defined.

如果某一点的导数不存在，则切线可能是竖直切线（极限存在）或根本不存在。在 IB 考试中，通常要求求导数存在点的切线方程。

2. General Equation of a Tangent Line | 切线方程的一般形式

For a point \( (x_1, y_1) \) on a curve \( y = f(x) \), the tangent line has slope \( m = f'(x_1) \). Its equation is:

对于曲线 \( y = f(x) \) 上的点 \( (x_1, y_1) \)，切线斜率为 \( m = f'(x_1) \)，其方程为：

y – y₁ = f'(x₁)(x – x₁)

This is the point-slope form of a straight line. It can also be rearranged into slope-intercept or general form depending on the question.

这是直线方程的点斜式，也可以根据题目要求改写为斜截式或一般式。

3. General Equation of a Normal Line | 法线方程的一般形式

The normal line at a point is perpendicular to the tangent line. Therefore its slope \( m_n \) satisfies the relation:

法线是过该点且与切线垂直的直线，因此其斜率 \( m_n \) 满足：

m_t × m_n = -1

Thus, for a tangent slope \( m_t = f'(x_1) \neq 0 \), the normal slope is:

因此，当切线斜率 \( m_t = f'(x_1) \neq 0 \) 时，法线斜率为：

m_n = -1 / f'(x₁)

If \( f'(x_1) = 0 \), the tangent is horizontal, so the normal is vertical (x = x₁). If the tangent is vertical, the normal is horizontal (y = y₁).

若 \( f'(x_1) = 0 \)，切线水平，则法线竖直，方程为 \( x = x_1 \)；若切线竖直，则法线水平，方程为 \( y = y_1 \)。

4. Steps to Find a Tangent Equation | 求切线方程的步骤

Follow these steps when asked to find the tangent line at a given point:

求给定点处的切线方程可遵循以下步骤：

Step 1: Differentiate \( y = f(x) \) to obtain \( f'(x) \).

第一步：对 \( y = f(x) \) 求导，得到 \( f'(x) \)。
Step 2: Substitute the x-coordinate of the given point into \( f'(x) \) to find the slope \( m \).

第二步：将给定点的横坐标代入 \( f'(x) \)，求出斜率 \( m \)。
Step 3: Use the point \( (x_1, y_1) \) and the slope \( m \) in the point-slope form.

第三步：利用点 \( (x_1, y_1) \) 和斜率 \( m \) 写出点斜式方程。
Step 4: Simplify if required (e.g. into \( y = mx + c \)).

第四步：按要求化简（例如化为 \( y = mx + c \) 的形式）。

5. Steps to Find a Normal Equation | 求法线方程的步骤

The normal line shares the same point but has a different slope. The steps are:

法线过同一点，但斜率不同，具体步骤如下：

Step 1: Find the tangent slope \( m_t = f'(x_1) \).

第一步：先求切线斜率 \( m_t = f'(x_1) \)。
Step 2: Compute \( m_n = -1/m_t \) if \( m_t \neq 0 \).

第二步：若 \( m_t \neq 0 \)，计算 \( m_n = -1/m_t \)。
Step 3: Write \( y – y_1 = m_n (x – x_1) \).

第三步：写出 \( y – y_1 = m_n (x – x_1) \)。
Step 4: Handle special cases: horizontal tangent gives vertical normal, etc.

第四步：处理特殊情况：切线水平则法线竖直，等等。

6. Example: Polynomial Curve | 例子：多项式曲线

Let \( f(x) = x^2 + 3x – 2 \). Find the tangent and normal at \( x = 1 \).

设 \( f(x) = x^2 + 3x – 2 \)，求该曲线在 \( x = 1 \) 处的切线与法线方程。

First find \( f(1) = 1 + 3 – 2 = 2 \). So the point is \( (1, 2) \).

先求 \( f(1) = 1 + 3 – 2 = 2 \)，因此点为 \( (1, 2) \)。

Differentiate: \( f'(x) = 2x + 3 \). At \( x = 1 \), \( m_t = 2(1) + 3 = 5 \).

求导得 \( f'(x) = 2x + 3 \)。在 \( x = 1 \) 处，\( m_t = 2(1) + 3 = 5 \)。

Tangent equation: \( y – 2 = 5(x – 1) \), i.e. \( y = 5x – 3 \).

切线方程：\( y – 2 = 5(x – 1) \)，即 \( y = 5x – 3 \)。

Normal slope: \( m_n = -1/5 \). Normal equation: \( y – 2 = -\frac{1}{5}(x – 1) \), or \( y = -\frac{1}{5}x + \frac{11}{5} \).

法线斜率 \( m_n = -1/5 \)。法线方程：\( y – 2 = -\frac{1}{5}(x – 1) \)，即 \( y = -\frac{1}{5}x + \frac{11}{5} \)。

Tangent: y = 5x – 3 , Normal: y = -(1/5)x + 11/5

7. Example: Trigonometric Function | 例子：三角函数

Consider \( y = \sin x \) at the point where \( x = \pi/3 \).

考虑 \( y = \sin x \) 在 \( x = \pi/3 \) 处的切线与法线。

We have \( f(\pi/3) = \sin(\pi/3) = \sqrt{3}/2 \). The derivative is \( f'(x) = \cos x \), so \( m_t = \cos(\pi/3) = 1/2 \).

因为 \( f(\pi/3) = \sin(\pi/3) = \sqrt{3}/2 \)。导数 \( f'(x) = \cos x \)，所以 \( m_t = \cos(\pi/3) = 1/2 \)。

Tangent: \( y – \sqrt{3}/2 = (1/2)(x – \pi/3) \).

切线：\( y – \sqrt{3}/2 = (1/2)(x – \pi/3) \)。

Normal slope: \( m_n = -2 \). Normal equation: \( y – \sqrt{3}/2 = -2(x – \pi/3) \).

法线斜率：\( m_n = -2 \)。法线方程：\( y – \sqrt{3}/2 = -2(x – \pi/3) \)。

Remember to use radian measure for trigonometric derivatives in IB.

注意：IB 中三角函数的导数必须使用弧度制。

8. Parametric Equations | 参数方程情形

When a curve is given in parametric form \( x = f(t), y = g(t) \), the slope of the tangent is found using the chain rule:

当曲线由参数方程 \( x = f(t), y = g(t) \) 给出时，切线斜率通过链式法则求得：

dy/dx = (dy/dt) / (dx/dt) , provided dx/dt ≠ 0

At a specific parameter value \( t_0 \), compute \( x(t_0) \), \( y(t_0) \), and the slope, then use the point-slope form as usual.

在特定参数值 \( t_0 \) 处，先计算 \( x(t_0) \)、\( y(t_0) \) 和斜率，然后照常使用点斜式。

For example, if \( x = t^2 \), \( y = t^3 \), then \( dy/dx = (3t^2)/(2t) = (3/2)t \) for \( t \neq 0 \). At \( t = 2 \), \( x = 4, y = 8 \), slope \( = 3 \). Tangent: \( y – 8 = 3(x – 4) \).

例如，若 \( x = t^2 \), \( y = t^3 \)，则 \( dy/dx = (3t^2)/(2t) = (3/2)t \)（\( t \neq 0 \)）。在 \( t = 2 \) 时，\( x = 4, y = 8 \)，斜率为 \( 3 \)。切线：\( y – 8 = 3(x – 4) \)。

9. Implicit Differentiation | 隐函数微分

For curves defined implicitly, such as \( x^2 + y^2 = 25 \), differentiate both sides with respect to \( x \), treating \( y \) as a function of \( x \).

对于隐式定义的曲线，如 \( x^2 + y^2 = 25 \)，对等式两边关于 \( x \) 求导，将 \( y \) 视为 \( x \) 的函数。

Differentiating gives \( 2x + 2y \frac{dy}{dx} = 0 \), so \( \frac{dy}{dx} = -x/y \). At a point such as \( (3, 4) \), the slope is \( -3/4 \).

求导得 \( 2x + 2y \frac{dy}{dx} = 0 \)，所以 \( \frac{dy}{dx} = -x/y \)。在点 \( (3, 4) \) 处，斜率为 \( -3/4 \)。

Tangent: \( y – 4 = -\frac{3}{4}(x – 3) \). Normal slope: \( 4/3 \). Normal: \( y – 4 = \frac{4}{3}(x – 3) \).

切线：\( y – 4 = -\frac{3}{4}(x – 3) \)。法线斜率为 \( 4/3 \)。法线：\( y – 4 = \frac{4}{3}(x – 3) \)。

Be careful when \( y = 0 \): the derivative may be undefined, often indicating a vertical tangent.

注意当 \( y = 0 \) 时导数可能无定义，这通常意味着存在竖直切线。

10. Common Mistakes and Pitfalls | 常见错误与陷阱

Students often make errors in the following areas:

学生在以下方面容易出错：

Using the original function value instead of the derivative value as the slope.

误将函数值当作斜率，而没有使用导数值。
Forgetting to apply the chain rule when differentiating composite functions.

对复合函数求导时忘记使用链式法则。
Assuming \( m_n = -m_t \) instead of \( m_n = -1/m_t \).

错误地认为 \( m_n = -m_t \)，而不是 \( m_n = -1/m_t \)。
Ignoring special cases where slopes are zero or infinite.

忽略斜率为零或无穷大的特殊情况。
Not checking that the given point actually lies on the curve.

未检查给定点是否真的在曲线上。

11. Workbook-Style Questions | 练习式问题

Try these typical IB-style questions to test your understanding:

尝试以下典型 IB 风格题目，检验你的理解：

Question	Hint
1. Find tangent to \( y = e^{2x} \) at \( x = 0 \).	Derivative is \( 2e^{2x} \); slope at 0 is 2.
2. Find normal to \( y = \ln x \) at \( x = 1 \).	Slope of tangent is 1; normal slope is -1.
3. For \( x^3 + y^3 = 9 \), find tangent at \( (1, 2) \).	Use implicit differentiation: \( dy/dx = -x^2/y^2 \).
4. Parametric curve \( x = t^2 – 1, y = t^2 + t \): find normal at \( t = 1 \).	Compute \( dy/dx = (2t+1)/(2t) \); at \( t=1 \), slope = 3/2.

12. Summary and Final Tips | 总结与最终建议

The key formulas for tangent and normal lines are based on the derivative. Always write the point-slope form carefully and convert to the required form.

切线与法线方程的核心公式都基于导数。务必准确写出点斜式，并按题目要求转换为相应形式。

In IB exams, show clear steps: differentiate, substitute, then form the equation. This maximizes method marks even if arithmetic errors occur.

在 IB 考试中，应展示清晰的步骤：求导、代入、写出方程。这样即使出现计算错误，也能获得方法分。

Practice with polynomials, trigonometric, exponential, logarithmic, parametric and implicit functions to be fully prepared.

建议练习多项式、三角函数、指数、对数、参数方程和隐函数等各类题型，以做好充分准备。

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Tangent and Normal Equations | 切线与法线方程