📚 Gradients of Curves | 曲线梯度
Understanding the gradient of a curve is the gateway to calculus. Straight lines have a constant slope, but curves change steepness at every point – finding that rate of change is exactly what differentiation delivers.
理解曲线梯度是微积分的入门关键。直线的斜率处处相同，但曲线上每一点的陡峭程度都在变化——求取这种变化率正是微分学的核心任务。
1. Why Gradients of Curves Matter | 曲线梯度的意义
In A-Level Mathematics, the gradient of a curve represents the instantaneous rate of change of one quantity with respect to another. It forms the basis for velocity–time graphs, optimisation problems, and modelling real-world behaviour.
在 A-Level 数学中，曲线梯度表示一个量相对于另一个量的瞬时变化率。它是速度–时间图像、最优化问题以及现实世界建模的基础。
Unlike a straight line, where you can pick any two points and get the same slope, a curve’s gradient is local – it differs from point to point. Understanding this local steepness is the first step toward the derivative.
与直线不同——直线上任意两点都能得到同一个斜率——曲线的梯度是局部的，每一点都不一样。理解这种局部陡峭程度是通往导数概念的第一步。
2. Refresher: Gradient of a Straight Line | 复习：直线的梯度
For two distinct points (x₁, y₁) and (x₂, y₂) on a straight line, the gradient m is the ratio of the vertical change to the horizontal change:
对于直线上的两个不同点 (x₁, y₁) 和 (x₂, y₂)，梯度 m 等于纵向变化与横向变化之比：
m = (y₂ − y₁) / (x₂ − x₁)
This value remains the same no matter which two points are chosen on the line. A positive m indicates an upward slope, a negative m a downward slope, and zero gives a horizontal line.
无论选取直线上哪两个点，这一数值都保持不变。m 为正表示向右上升，为负表示下降，为零时则是水平线。
3. The Problem with Curves | 曲线带来的难题
If you try to apply the same rise-over-run formula to a curve, you get an average gradient over an interval rather than the exact steepness at a single point. Different chords on the same curve give different slopes.
如果试图将同样的"竖直除以水平"公式直接用于曲线，你得到的只是一个区间上的平均梯度，而不是某一点的准确陡峭程度。同一条曲线上的不同弦会给出不同的斜率。
We need a way to zoom in so closely that the curve looks straight, allowing us to measure the gradient exactly where we want it.
我们需要一种方法，把视野放大到曲线看起来像直线的程度，从而能在我们想要的点精确测出梯度。
4. Using Secants and Chords | 利用割线与弦
A secant line is a straight line through two points on a curve. Its gradient approximates the curve’s average rate of change between those points.
割线是通过曲线上两点的直线。它的梯度近似表示曲线在两点之间的平均变化率。
of (1.44 − 1) / (1.2 − 1) = 0.44 / 0.2 = 2.2. This is the average gradient over [1, 1.2].
以 f(x) = x² 为例，取两点 A(1, 1) 与 B(1.2, 1.44)。弦 AB 的梯度为 (1.44 − 1) / (1.2 − 1) = 0.44 / 0.2 = 2.2。这是 [1, 1.2] 区间上的平均梯度。
To refine the estimate, we bring B closer to A. The chord gradually turns into a line that just touches the curve at A – a tangent.
为了改进估计，我们将 B 向 A 靠拢。弦逐渐转变成一条仅与曲线在 A 点接触的直线——切线。
5. Moving Closer: From Chord to Tangent | 逐步逼近：从弦到切线
Let B have x-coordinate 1 + h, where h is a small number. The gradient of the chord AB is:
令 B 的横坐标为 1 + h，h 是一个很小的数。弦 AB 的梯度为：
(f(1 + h) − f(1)) / h
The table below shows how this gradient behaves for f(x) = x² as h gets smaller.
下表展示了 f(x) = x² 的情形下，随着 h 减小这一梯度的变化趋势。
| h | Chord gradient ((1+h)² − 1) / h |
| 0.1 | 2.1 |
| 0.01 | 2.01 |
| 0.001 | 2.001 |
| 0.0001 | 2.0001 |
adient approaches 2. The limiting value is precisely the gradient of the tangent at x = 1.
当 h 趋近于 0 时，弦的梯度趋近于 2。这个极限值正是 x = 1 处切线的梯度。
6. The Limit Definition of a Gradient Function | 梯度函数的极限定义
ient at any point x, we write h for the small horizontal step and take the limit as h → 0:
为求任意点 x 处的梯度，我们用 h 表示微小的水平步长，并令 h → 0 取极限：
f'(x) = limh→0 (f(x + h) − f(x)) / h
This expression is called the derivative of f(x) and gives the gradient function of the curve. The notation f'(x) is read as "f prime of x".
这一表达式称为 f(x) 的导数，它给出了曲线的梯度函数。记号 f'(x) 读作 "f prime of x"。
7. Differentiation from First Principles | 从第一原理求导
fferentiation from first principles. Let us find the derivative of f(x) = x² step by step.
直接使用极限定义求导叫作从第一原理求导。我们逐步求出 f(x) = x² 的导数。
Write f(x + h) = (x + h)² = x² + 2xh + h².
写出 f(x + h) = (x + h)² = x² + 2xh + h²。
ence quotient: (f(x + h) − f(x)) / h = (x² + 2xh + h² − x²) / h = (2xh + h²) / h = 2x + h.
构造差商：(f(x + h) − f(x)) / h = (x² + 2xh + h² − x²) / h = (2xh + h²) / h = 2x + h。
Take the limit as h → 0: limh→0 (2x + h) = 2x.
取 h → 0 时的极限：limh→0 (2x + h) = 2x。
Therefore, the gradient function for f(x) = x² is f'(x) = 2x. At x = 3, the gradient is 6; at x = −1, it is −2.
因此，f(x) = x² 的梯度函数是 f'(x) = 2x。在 x = 3 处梯度为 6；在 x = −1 处为 −2。
8. Gradient Functions for Simple Polynomials | 简单多项式的梯度函数
pplying first principles to f(x) = x³ gives (x + h)³ = x³ + 3x²h + 3xh² + h³, leading to f'(x) = 3x². A clear pattern emerges, often called the power rule.
对 f(x) = x³ 使用第一原理，有 (x + h)³ = x³ + 3x²h + 3xh² + h³，得出 f'(x) = 3x²。一个清晰的模式出现了，通常称为幂法则。
For any f(x) = xⁿ, where n is a constant, the derivative is:
对任意 f(x) = xⁿ（n 为常数），导数为：
f'(x) = n xⁿ⁻¹
If f(x) = x, then f'(x) = 1 × x⁰ = 1.
若 f(x) = x，则 f'(x) = 1 × x⁰ = 1。
If f(x) = x⁴, then f'(x) = 4x³.
若 f(x) = x⁴，则 f'(x) = 4x³。
If f(x) = 5x², the constant multiple rule gives f'(x) = 5 × 2x = 10x.
若 f(x) = 5x²，根据常数倍法则，f'(x) = 5 × 2x = 10x。
When a power is negative or fractional, such as f(x) = x⁻² or f(x) = x½, the rule still applies: multiply by the power and reduce the power by one.
当指数为负数或分数时，如 f(x) = x⁻² 或 f(x) = x½，法则依然适用：指数乘以系数，指数再减一。
9. Notation for the Derivative | 导数的记法
There are several common notations for the gradient function, all meaning the same thing.
梯度函数有几种常用记法，含义完全相同。
| Notation | Meaning |
| f'(x) | Derivative of f with respect to x |
| dy/dx | Derivative of y with respect to x |
| d/dx (expression) | The operator that finds the derivative |
In Edexcel exams, you should be comfortable switching between f'(x) and dy/dx. For example, if y = 2x³ − 4x + 7, you can write dy/dx = 6x² − 4.
在 Edexcel 考试中，你需要自如地在 f'(x) 和 dy/dx 之间切换。例如，若 y = 2x³ − 4x + 7，你可以写成 dy/dx = 6x² − 4。
10. Finding Tangents and Normals | 求切线方程与法线方程
Once you know the gradient at a point, you can write the equation of the tangent line. If a curve passes through (a, f(a)) and has gradient m there, the tangent is:
知道某点的梯度之后，你就可以写出切线的方程。若曲线经过 (a, f(a)) 且此处的梯度为 m，则切线方程为：
y − f(a) = m (x − a)
The normal is perpendicular to the tangent. Its gradient is −1/m (provided m ≠ 0), and its equation follows the same point–gradient form.
法线与切线垂直。其梯度为 −1/m（只要 m ≠ 0），并同样遵循点–斜式方程。
s y − 9 = 6(x − 3) → y = 6x − 9. The normal has gradient −1/6, giving y − 9 = −1/6(x − 3).
例如，考察 y = x² 在点 (3, 9) 处。梯度为 6，因此切线为 y − 9 = 6(x − 3) → y = 6x − 9。法线梯度为 −1/6，得到 y − 9 = −1/6(x − 3)。
11. Gradients as Rates of Change in Context | 上下文中的梯度即变化率
In real-world problems, a gradient can represent velocity, acceleration, rate of growth, or marginal cost. For a displacement–time graph, the gradient dy/dx is the velocity; for a velocity–time graph, the gradient dv/dt is the acceleration.
在现实问题中，梯度可以代表速度、加速度、增长率或边际成本。对于位移–时间图像，梯度 dy/dx 表示速度；对于速度–时间图像，梯度 dv/dt 表示加速度。
This interpretation makes the concept of gradient of a curve incredibly powerful across physics, economics, and biology.
这种解释使曲线梯度的概念在物理、经济和生物等领域具有极强的威力。
Whenever a context asks for "the rate of change of P with respect to t at t = 5", you are being asked for the derivative dP/dt evaluated at t = 5 – exactly the gradient of the curve at that instant.
只要题目询问"在 t = 5 时 P 关于 t 的变化率"，你实际上就是在求导数 dP/dt 在 t = 5 处的值——正是曲线在该瞬间的梯度。
12. Key Takeaways | 关键要点总结
The gradient of a curve at a point is the gradient of its tangent there.
曲线在某点的梯度就是该点切线的梯度。
The limit (f(x + h) − f(x)) / h as h → 0 defines the derivative f'(x) or dy/dx.
h → 0 时 (f(x + h) − f(x)) / h 的极限定义了导数 f'(x) 或 dy/dx。
For f(x) = xⁿ, the derivative is f'(x) = n xⁿ⁻¹.
对于 f(x) = xⁿ，导数为 f'(x) = n xⁿ⁻¹。
Tangent equation: y − y₁ = m (x − x₁). Normal gradient: −1/m.
切线方程：y − y₁ = m (x − x₁)。法线梯度：−1/m。
Gradients represent instantaneous rates of change in practical contexts.
梯度在实践情境中代表瞬时变化率。
Mastering the gradient of curves sets you up for the whole of differentiation, from stationary points to optimisation and beyond.
掌握曲线梯度能让你从容面对整个微分学，从驻点到最优化问题，乃至更远的内容。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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