📚 Parametric Equations and Differentiation | 参数方程与求导
Parametric equations describe a curve by expressing both x and y in terms of a third variable, usually t or θ. In Edexcel A-Level Mathematics this topic links algebra, coordinate geometry, trigonometry and calculus, so it is a favourite area for examiners to combine several skills inside a single question.
参数方程通过用第三个变量（通常是 t 或 θ）同时表示 x 与 y 来描述曲线。在 Edexcel A-Level 数学中，这一主题把代数、坐标几何、三角与微积分串联起来，因此考官非常喜欢在一个题目里同时考查多项技能。
This guide covers the definition of parametric equations, conversion to Cartesian form, differentiation using the chain rule, second derivatives, tangents and normals, stationary points, curve sketching and exam technique. Every formula is written with Unicode symbols so that it displays correctly on any device.
本文涵盖参数方程的定义、化为直角坐标式、用链式法则求导、二阶导数、切线与法线、驻点、曲线作图以及考试技巧。所有公式均使用 Unicode 符号书写，以便在任何设备上正确显示。
1. Understanding Parametric Equations | 理解参数方程
A parametric curve is defined by two equations, x = f(t) and y = g(t), where t is the parameter. As t varies over its allowed interval, the point (x, y) traces out a curve in the xy-plane. The parameter itself is not usually plotted; it simply controls the position of the point.
参数曲线由两个方程定义：x = f(t) 和 y = g(t)，其中 t 为参数。当 t 在其允许区间内变化时，点 (x, y) 在 xy 平面上勾勒出一条曲线。参数本身通常不出现在图像上，它只是控制点的位置。
For example, x = t² and y = 2t represent the parabola y² = 4x the same curve as the Cartesian parabola, but the parametric version also tells you how the curve is traced: as t increases from negative values to positive values, the point moves from the lower branch to the upper branch.
例如，x = t² 与 y = 2t 在消去 t 后表示抛物线 y² = 4x，因为 t = y ÷ 2，所以 x = (y ÷ 2)² = y² ÷ 4。它与直角坐标形式的抛物线是同一条曲线，但参数形式还告诉你曲线是如何被描绘的：当 t 从负值增大到正值时，点从下分支移动到上分支。
Parametric representations are especially useful when a curve is not a single-valued function, when a curve loops back on itself, or when modelling motion where t represents time. In mechanics-style questions, x(t) and y(t) are the horizontal and vertical positions of a particle, and the parameter t has a clear physical meaning.
当曲线不是单值函数、当曲线形成回环，或者用 t 表示时间进行运动建模时，参数表示尤其有用。在力学类问题中，x(t) 与 y(t) 分别是质点的水平与竖直位置，参数 t 具有明确的物理意义。
Key vocabulary you should know:
你需要掌握的关键词汇：
- Parameter: the controlling variable, often t or θ. 参数：起控制作用的变量，通常是 t 或 θ。
- Parametric form: the pair x = f(t), y = g(t). 参数形式：方程组 x = f(t)、y = g(t)。
- Cartesian form: a single equation in x and y only. 直角坐标形式：只含 x 与 y 的单一方程。
- Orientation: the direction in which t increases along the curve. 方向：t 增大时曲线被描绘的方向。
2. Converting Between Parametric and Cartesian Forms | 参数式与直角坐标式的互化
To find the Cartesian equation, you must eliminate the parameter t from the two equations. The standard strategies are substitution, rearranging one equation for t and substituting into the other, and using trigonometric identities such as sin²θ + cos²θ = 1.
要找到直角坐标方程，你需要从两个方程中消去参数 t。常用方法包括代入法、从其中一个方程解出 t 再代入另一个方程，以及使用三角恒等式，例如 sin²θ + cos²θ = 1。
Method 1: Substitution. riction on the domain is needed here, because t can be any real number and x = t + 1 also covers all real numbers.
方法一：代入法。若 x = t + 1 且 y = 2t - 3，写出 t = x - 1，再代入 y：y = 2(x - 1) - 3 = 2x - 5。直角坐标方程为直线 y = 2x - 5。这里不需要对定义域加限制，因为 t 可取任意实数，且 x = t + 1 也能取遍所有实数。
Method 2: Trigonometric identity. 9. This is a circle of radius 3 centred at the origin, traced once as θ goes from 0 to 2π.
方法二：三角恒等式。若 x = 3cosθ 且 y = 3sinθ，则 x ÷ 3 = cosθ，y ÷ 3 = sinθ。利用 cos²θ + sin²θ = 1 得 (x ÷ 3)² + (y ÷ 3)² = 1，即 x² + y² = 9。这是以原点为圆心、半径为 3 的圆，当 θ 从 0 变到 2π 时恰好描绘一周。
Method 3: Rearranging products. ular hyperbola, and the parametric form is often much easier to differentiate than the Cartesian form.
方法三：整理乘积关系。若 x = 2t 且 y = 4 ÷ t，则 t = x ÷ 2，y = 4 ÷ (x ÷ 2) = 8 ÷ x，所以 xy = 8。这是一条直角双曲线，而参数形式通常比直角坐标形式更容易求导。
The table below summarises the identities you will use most often in Edexcel questions:
下表总结了 Edexcel 题目中最常用的恒等式：
| Parametric form 参数形式 | Identity used 所用恒等式 | Cartesian form 直角坐标式 |
|---|---|---|
| x = a cosθ, y = a sinθ | cos²θ + sin²θ = 1 | x² + y² = a² |
| x = a cosθ, y = b sinθ | cos²θ + sin²θ = 1 | x² ÷ a² + y² ÷ b² = 1 |
| x = a secθ, y = b tanθ | sec²θ - tan²θ = 1 | x² ÷ a² - y² ÷ b² = 1 |
| x = t + 1 ÷ t, y = t - 1 ÷ t | (a + b)² - (a - b)² = 4ab | x² - y² = 4 |
3. Domains, Ranges and Restrictions on the Parameter | 参数的定义域、值域与限制
0, then x ≥ 0 and y ≥ 0, so only the upper-right half of the parabola y² = 4x is traced. Examiners often award a mark for stating such a restriction.
参数定义并不总是等价于整条直角坐标曲线，因为 t 的取值范围可能只覆盖其中一部分。若 x = t²、y = 2t 且 t ≥ 0，则 x ≥ 0、y ≥ 0，因此只能描绘出抛物线 y² = 4x 的右上半部分。考官常常为写出这类限制单独给分。
When you convert to Cartesian form, always ask two questions: which x-values are possible, and which y-values are possible? A quick sketch or a table of values for t will usually answer both.
当你化为直角坐标式时，务必问两个问题：x 可以取哪些值？y 可以取哪些值？快速画草图或列出 t 的取值表通常就能回答这两个问题。
e same point, the curve may cross itself or retrace its path, and you should mention this if a question asks about the shape or the number of intersections.
还要记住，参数曲线上的同一个点可能由多个 t 值得到，对称曲线尤其如此。如果 t = 1 与 t = -1 给出同一个点，曲线可能自交或重复描绘，若题目问及形状或交点个数，你应当指出这一点。
4. Differentiating Parametric Equations: The Chain Rule | 参数方程求导：链式法则
t) × (dt ÷ dx). Because dt ÷ dx is the reciprocal of dx ÷ dt, the working formula becomes:
本主题的核心结论来自链式法则。由于 y 依赖于 t，而 t 依赖于 x，我们有 dy ÷ dx = (dy ÷ dt) × (dt ÷ dx)。又因为 dt ÷ dx 是 dx ÷ dt 的倒数，于是得到实用公式：
dy ÷ dx = (dy ÷ dt) ÷ (dx ÷ dt)
vertical, and dy ÷ dx is undefined there. You should state this condition when it matters.
只要在所讨论的点处 dx ÷ dt ≠ 0，该公式就成立。若 dx ÷ dt = 0 而 dy ÷ dt ≠ 0，则切线竖直，此时 dy ÷ dx 不存在。在相关情形下，你应当写明这一条件。
The practical procedure is always the same:
实际操作步骤始终相同：
- Differentiate x with respect to t to obtain dx ÷ dt. 对 x 关于 t 求导，得到 dx ÷ dt。
- Differentiate y with respect to t to obtain dy ÷ dt. 对 y 关于 t 求导，得到 dy ÷ dt。
- Divide dy ÷ dt by dx ÷ dt, and simplify fully. 用 dy ÷ dt 除以 dx ÷ dt，并彻底化简。
- Substitute the given value of t to find the gradient at that point. 代入给定的 t 值，求出该点处的斜率。
5. Worked Example: Finding dy/dx Step by Step | 例题精讲：逐步求 dy/dx
Example 1. ons x = t² + 1 and y = t³ - 3t. Find dy ÷ dx in terms of t, and find its value when t = 2.
例 1。曲线参数方程为 x = t² + 1 与 y = t³ - 3t。求用 t 表示的 dy ÷ dx，并求 t = 2 时的值。
= 2.25. The point on the curve at t = 2 is (5, 2), so the tangent there has gradient 2.25.
分别对 t 求导：dx ÷ dt = 2t，dy ÷ dt = 3t² - 3。因此 dy ÷ dx = (3t² - 3) ÷ (2t)。当 t = 2 时，dy ÷ dx = (12 - 3) ÷ 4 = 9 ÷ 4 = 2.25。t = 2 时曲线上的点为 (5, 2)，故该点切线斜率为 2.25。
Example 2. A curve has x = 2cosθ and y = 3sinθ. Find dy ÷ dx in terms of θ.
例 2。曲线参数方程为 x = 2cosθ 与 y = 3sinθ。求用 θ 表示的 dy ÷ dx。
inθ) = -(3 ÷ 2)cotθ. At θ = π ÷ 4, the gradient is -(3 ÷ 2) × 1 = -1.5, and the point is (√2, 3√2 ÷ 2).
这里 dx ÷ dθ = -2sinθ，dy ÷ dθ = 3cosθ，所以 dy ÷ dx = (3cosθ) ÷ (-2sinθ) = -(3 ÷ 2)cotθ。当 θ = π ÷ 4 时，斜率为 -(3 ÷ 2) × 1 = -1.5，对应点为 (√2, 3√2 ÷ 2)。
Example 3. A curve has x = t + 1 ÷ t and y = t - 1 ÷ t, with t ≠ 0. Find dy ÷ dx.
例 3。曲线参数方程为 x = t + 1 ÷ t 与 y = t - 1 ÷ t，且 t ≠ 0。求 dy ÷ dx。
ding gives dy ÷ dx = (1 + 1 ÷ t²) ÷ (1 - 1 ÷ t²) = (t² + 1) ÷ (t² - 1), valid when t ≠ ±1.
dx ÷ dt = 1 - 1 ÷ t²，dy ÷ dt = 1 + 1 ÷ t²。相除得 dy ÷ dx = (1 + 1 ÷ t²) ÷ (1 - 1 ÷ t²) = (t² + 1) ÷ (t² - 1)，在 t ≠ ±1 时成立。
6. The Second Derivative d²y/dx² | 二阶导数 d²y/dx²
The second derivative is found by differentiating dy ÷ dx with respect to x, not with respect to t. Applying the chain rule a second time gives:
二阶导数是对 dy ÷ dx 关于 x 求导得到的，而不是关于 t 求导。再次使用链式法则可得：
d²y ÷ dx² = (d ÷ dt)(dy ÷ dx) ÷ (dx ÷ dt)
s wrong. You must differentiate the first derivative with respect to t, then divide the result by dx ÷ dt once more.
考试中非常常见的错误是把 d²y ÷ dx² 写成 (d²y ÷ dt²) ÷ (d²x ÷ dt²)。这是错误的。你必须先对 t 求一阶导数的导数，然后再除以一次 dx ÷ dt。
Worked example. For x = t² and y = t³, find d²y ÷ dx².
例题。已知 x = t²、y = t³，求 d²y ÷ dx²。
y divide by dx ÷ dt = 2t to obtain d²y ÷ dx² = (3 ÷ 2) ÷ (2t) = 3 ÷ (4t), valid for t ≠ 0.
dx ÷ dt = 2t，dy ÷ dt = 3t²，因此 dy ÷ dx = 3t² ÷ 2t = 3t ÷ 2。再对 t 求导：d ÷ dt(3t ÷ 2) = 3 ÷ 2。最后除以 dx ÷ dt = 2t，得 d²y ÷ dx² = (3 ÷ 2) ÷ (2t) = 3 ÷ (4t)，在 t ≠ 0 时成立。
Second example. le gives (3t² + 3) ÷ (2t²). Dividing by dx ÷ dt = 2t yields d²y ÷ dx² = 3(t² + 1) ÷ (4t³).
第二个例子。已知 x = t² + 1、y = t³ - 3t，则 dy ÷ dx = (3t² - 3) ÷ (2t)。用商法则对 t 求导得 (3t² + 3) ÷ (2t²)。再除以 dx ÷ dt = 2t，得 d²y ÷ dx² = 3(t² + 1) ÷ (4t³)。
The sign of d²y ÷ dx² tells you the concavity of the curve: positive means concave up, negative means concave down. This is useful when deciding whether a stationary point is a maximum or a minimum.
d²y ÷ dx² 的符号反映曲线的凹凸性：为正表示上凹，为负表示下凹。判断驻点是极大值还是极小值时非常有用。
7. Tangents and Normals to Parametric Curves | 参数曲线的切线与法线
x - x₁). The normal at the same point has gradient -1 ÷ m, provided that m ≠ 0, and satisfies y - y₁ = (-1 ÷ m)(x - x₁).
求参数曲线在给定点处的切线时，先在该 t 值处求出 x、y 与 dy ÷ dx，再用直线公式 y - y₁ = m(x - x₁)。同一点处的法线斜率为 -1 ÷ m（要求 m ≠ 0），满足 y - y₁ = (-1 ÷ m)(x - x₁)。
Example. is y - 2 = -1(x - 1), that is y = -x + 3. These two lines meet at (1, 2) at right angles.
例题。曲线 x = t²、y = 2t 的 dy ÷ dx = 2 ÷ (2t) = 1 ÷ t。当 t = 1 时点为 (1, 2)，斜率为 1。切线为 y - 2 = 1(x - 1)，即 y = x + 1。法线为 y - 2 = -1(x - 1)，即 y = -x + 3。两直线在 (1, 2) 处垂直相交。
uestion asks where the tangent is parallel to a given line, set dy ÷ dx equal to that line' y-axis, you need dx ÷ dt = 0 with dy ÷ dt ≠ 0, and the tangent equation is simply x = x₁.
若题目问切线何时与给定直线平行，则令 dy ÷ dx 等于该直线的斜率并解出 t。若切线与 y 轴平行，则需要 dx ÷ dt = 0 且 dy ÷ dt ≠ 0，此时切线方程就是 x = x₁。
8. Stationary Points and Curve Sketch
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