A-Level Mathematics: Tangents, Normals and Gradients of Curves | A-Level数学：曲线的切线、法线与梯度

📚 A-Level Mathematics: Tangents, Normals and Gradients of Curves | A-Level数学：曲线的切线、法线与梯度

The gradient of a curve at a given point is a fundamental idea in calculus. Unlike a straight line, whose gradient is constant, a curve changes direction continuously. Differentiation provides a powerful tool for measuring this rate of change and for writing down the equations of the tangent and normal lines that touch or cross the curve at a particular point.

曲线在某一点的梯度是微积分中的基本概念。与斜率恒定的直线不同，曲线会持续改变方向。微分提供了测量这种变化率的强大工具，也可以用来写出在特定点与曲线相切或垂直的切线方程与法线方程。

1. The Gradient Function: f'(x) and dy/dx | 1. 梯度函数：f'(x) 与 dy/dx

For a curve defined by y = f(x), the derivative f'(x), also written as dy/dx, gives the gradient of the curve at any point x. It is a function that describes how steeply the curve is rising or falling at each value of x.

对于由 y = f(x) 定义的曲线，导数 f'(x)（也写作 dy/dx）给出曲线在任意点 x 处的梯度。它是一个函数，描述曲线在每个 x 值处上升或下降的陡峭程度。

The rule for differentiating a power function is:

幂函数的微分法则为：

If y = xⁿ, then dy/dx = n·xⁿ⁻¹

若 y = xⁿ，则 dy/dx = n·xⁿ⁻¹

Constants have derivative zero, and for a constant multiple c·xⁿ the derivative is c·n·xⁿ⁻¹.

常数的导数为零；对于常数倍 c·xⁿ，其导数为 c·n·xⁿ⁻¹。

2. Evaluating the Gradient at a Point | 2. 计算某一点的梯度

To find the gradient of a curve at a specific point x = a, substitute x = a into the gradient function f'(x). The result is the slope m of the tangent line at that point.

要求曲线在特定点 x = a 处的梯度，只需将 x = a 代入梯度函数 f'(x)。所得结果即为该点切线 l 的斜率 m。

For example, y = x² + 3x − 5 gives dy/dx = 2x + 3. At x = 1, m = 2(1) + 3 = 5.

例如，y = x² + 3x − 5 的导数为 dy/dx = 2x + 3。在 x = 1 处，m = 2(1) + 3 = 5。

If the derivative is zero at a point, the tangent is horizontal. If the derivative is undefined, the tangent may be vertical.

如果导数在某点为 0，则切线水平。如果导数无定义，则切线可能垂直。

3. Equation of the Tangent | 3. 切线方程

Once the gradient m at x = a is known, the equation of the tangent line can be found using the point-slope form of a straight line. If the point on the curve is (a, f(a)), the tangent equation is:

一旦求出 x = a 处的梯度 m，便可用直线的点斜式写出切线方程。若曲线上的点为 (a, f(a))，切线方程为：

y − f(a) = m(x − a)

Worked example: For y = x³ − 2x at x = 2, first y = 8 − 4 = 4, and dy/dx = 3x² − 2, so m = 3(4) − 2 = 10. The tangent is y − 4 = 10(x − 2), which simplifies to y = 10x − 16.

例题：对于 y = x³ − 2x，在 x = 2 处，先算 y = 8 − 4 = 4；dy/dx = 3x² − 2，所以 m = 3(4) − 2 = 10。切线为 y − 4 = 10(x − 2)，化简得 y = 10x − 16。

4. Equation of the Normal | 4. 法线方程

The normal to a curve at a point is the straight line perpendicular to the tangent at that point. If the tangent has gradient m ≠ 0, the normal has gradient −1/m.

曲线在某点的法线，是在该点与切线垂直的直线。若切线的梯度为 m ≠ 0，则法线的梯度为 −1/m。

m_normal × m_tangent = −1

m_法线 × m_切线 = −1

Using the same point (a, f(a)), the normal equation is:

使用同一点 (a, f(a))，法线方程为：

y − f(a) = (−1/m)(x − a)

Special case: if m = 0, the tangent is horizontal and the normal is vertical, with equation x = a.

特殊情况：若 m = 0，则切线水平，法线垂直，方程为 x = a。

5. Worked Example: Tangent and Normal Together | 5. 综合例题：切线与法线

Consider y = √x at x = 4. At x = 4, y = 2, so the point is (4, 2). Write y = x^(1/2), then dy/dx = (1/2)x^(−1/2) = 1/(2√x). At x = 4, m = 1/(2×2) = 1/4.

考虑 y = √x 在 x = 4 处。当 x = 4 时，y = 2，因此点为 (4, 2)。将 y 写成 x^(1/2)，则 dy/dx = (1/2)x^(−1/2) = 1/(2√x)。在 x = 4 处，m = 1/(2×2) = 1/4。

Tangent: y − 2 = (1/4)(x − 4), giving y = (1/4)x + 1.

切线：y − 2 = (1/4)(x − 4)，即 y = (1/4)x + 1。

Normal gradient = −4. Normal: y − 2 = −4(x − 4), giving y = −4x + 18.

法线梯度为 −4。法线：y − 2 = −4(x − 4)，即 y = −4x + 18。

6. Parametric Curves: Gradients and Tangents | 6. 参数曲线：梯度与切线

For a curve defined parametrically by x = g(t) and y = h(t), the gradient is found by dividing the derivative of y with respect to t by the derivative of x with respect to t:

对于由 x = g(t) 和 y = h(t) 定义的参数曲线，梯度通过将 y 对 t 的导数除以 x 对 t 的导数得到：

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

dy/dx = (dy/dt) ÷ (dx/dt)，其中 dx/dt ≠ 0

For example, x = t², y = 2t + 1. Then dx/dt = 2t and dy/dt = 2, so dy/dx = 2/(2t) = 1/t. At t = 3, the gradient is 1/3.

例如，x = t²，y = 2t + 1。则 dx/dt = 2t，dy/dt = 2，所以 dy/dx = 2/(2t) = 1/t。在 t = 3 处，梯度为 1/3。

7. Implicit Differentiation and Normals | 7. 隐函数微分与法线

When a curve is given implicitly, such as x² + y² = 25, differentiate every term with respect to x, remembering that y is a function of x. The chain rule gives d/dx(y²) = 2y·dy/dx.

当曲线以隐式给出时，例如 x² + y² = 25，对每一项关于 x 求导，并记住 y 是 x 的函数。链式法则给出 d/dx(y²) = 2y·dy/dx。

For x² + y² = 25: 2x + 2y·dy/dx = 0, so dy/dx = −x/y.

对于 x² + y² = 25：2x + 2y·dy/dx = 0，因此 dy/dx = −x/y。

At the point (3, 4), m = −3/4. The tangent equation is y − 4 = (−3/4)(x − 3), and the normal gradient is 4/3, giving the normal y − 4 = (4/3)(x − 3).

在点 (3, 4) 处，m = −3/4。切线方程为 y − 4 = (−3/4)(x − 3)，而法线梯度为 4/3，法线为 y − 4 = (4/3)(x − 3)。

8. Increasing and Decreasing Functions | 8. 函数的递增与递减

The sign of the gradient function tells us how the curve behaves over an interval:

梯度函数的符号告诉我们曲线在区间上的行为：

Condition 条件	Meaning 含义
f'(x) > 0	The function is increasing 函数递增
f'(x) < 0	The function is decreasing 函数递减
f'(x) = 0	Horizontal tangent; possible stationary point 切线水平；可能是驻点

These conditions are essential for sketching curves and locating maximum and minimum points.

这些条件对于绘制曲线图像以及寻找极大值和极小值点至关重要。

9. Stationary Points and Tangent Gradients | 9. 驻点与切线梯度

A stationary point occurs where dy/dx = 0. At such a point the tangent is horizontal, so its equation is simply y = f(a).

驻点出现在 dy/dx = 0 处。在该点切线水平，因此切线方程简化为 y = f(a)。

For y = x² − 4x + 7, dy/dx = 2x − 4 = 0 gives x = 2. Then y = 4 − 8 + 7 = 3, so the stationary point is (2, 3) and the horizontal tangent is y = 3.

对于 y = x² − 4x + 7，dy/dx = 2x − 4 = 0，解得 x = 2。于是 y = 4 − 8 + 7 = 3，所以驻点为 (2, 3)，水平切线为 y = 3。

To classify the stationary point, use the second derivative or test the sign of dy/dx on either side.

要判断驻点的类型，可以使用二阶导数，或者检验 dy/dx 在两侧的符号。

10. Common Pitfalls and Exam Tips | 10. 常见错误与考试技巧

When writing equations of tangents and normals, students often confuse the two formulas. Always ask: does this line touch the curve, or is it perpendicular to the touch line?

在写切线与法线方程时，学生常混淆两个公式。要始终问自己：这条线是接触曲线，还是与接触线垂直？

Never divide by zero when finding the normal gradient; use x = a for vertical lines.

求法线梯度时不能除以 0；若切线水平，法线垂直，方程为 x = a。
Simplify the final equation into the required form, often y = mx + c or ax + by + c = 0.

将最终方程化简为题目要求的形式，通常是 y = mx + c 或 ax + by + c = 0。
For parametric equations, check whether the corresponding point satisfies the given parameter value.

对于参数方程，先确认给定参数值对应的点是否满足题意。

11. Summary of Key Formulas | 11. 关键公式总结

The table below gathers the central results for quick revision:

下表汇总了核心结论，方便快速复习：

Concept 概念	Formula 公式
Gradient at x = a 在 x = a 处的梯度	m = f'(a)
Tangent 切线	y − f(a) = f'(a)(x − a)
Normal 法线	y − f(a) = (−1/f'(a))(x − a)
Parametric gradient 参数梯度	dy/dx = (dy/dt)/(dx/dt)

Mastering these relationships between a curve and its straight-line companions will build a strong foundation for optimisation, kinematics and further calculus.

掌握曲线与其直线伙伴——切线与法线——之间的关系，将为最优化、运动学以及更深入的微积分打下坚实基础。

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A-Level Mathematics: Tangents, Normals and Gradients of Curves | A-Level数学：曲线的切线、法线与梯度