Parametric Equations: Differentiation and Integration | 参数方程：求导与积分

📚 Parametric Equations: Differentiation and Integration | 参数方程：求导与积分

In Edexcel A-Level Pure Mathematics, parametric equations are used to describe curves where both x and y depend on a third variable t. This topic appears regularly in Paper 1 and Paper 2, often combining coordinate geometry, differentiation, and integration. A strong understanding of parametric methods is essential for exam success, especially in questions on tangents, normals, stationary points, and areas under curves.

在 Edexcel A-Level 纯数学中，参数方程用来描述 x 和 y 都依赖于第三个变量 t 的曲线。这一主题经常出现在 Paper 1 和 Paper 2 中，常与坐标几何、微分和积分结合考查。扎实掌握参数方法对于考试成功至关重要，尤其是在切线、法线、驻点和曲线下面积等问题中。

1. Parametric Basics | 参数方程基本概念

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 domain, the point (x, y) traces out a curve in the Cartesian plane. The parameter often represents time, an angle, or another quantity that 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。随着 t 在其定义域内变化，点 (x, y) 在笛卡尔平面内描绘出一条曲线。参数通常表示时间、角度或其他控制点位置的量。

In Edexcel questions, the parameter t may be restricted to a given interval, such as t ∈ [-2, 2] or θ ∈ [0, 2π]. The restriction determines which part of the curve is being considered, so it is important to use the given domain when plotting or calculating.

在 Edexcel 考题中，参数 t 可能被限制在给定区间内，例如 t ∈ [-2, 2] 或 θ ∈ [0, 2π]。这个限制决定了曲线被考虑的部分，因此在绘图或计算时必须使用给定的定义域。

2. Why Use Parameters? | 为什么使用参数方程

Some curves have much simpler parametric forms than Cartesian equations. For example, a circle of radius r centred at the origin can be written as x = r cos θ, y = r sin θ, where θ is the angle measured from the positive x-axis. This form is often easier to differentiate and integrate when working with geometrical problems.

有些曲线的参数形式比笛卡尔方程简单得多。例如，以原点为圆心、半径为 r 的圆可以写成 x = r cos θ，y = r sin θ，其中 θ 是从正 x 轴量起的角度。在处理几何问题时，这种形式通常更容易求导和积分。

Circle: x = r cos θ, y = r sin θ
Ellipse: x = a cos θ, y = b sin θ
Parabola: x = at², y = 2at

The parameter form also allows us to describe motion along a curve more naturally, since x and y can be viewed as functions of time t. This makes parametric equations useful in applied contexts and in modelling real-world paths.

参数形式还能更自然地描述沿曲线的运动，因为 x 和 y 可以看作时间 t 的函数。这使得参数方程在应用情境和实际路径建模中非常有用。

3. First Derivative: Gradient of a Parametric Curve | 一阶导数：参数曲线的斜率

To find the gradient of a parametric curve, we cannot directly differentiate y with respect to x. Instead, we use the chain rule to write dy/dx as the ratio of dy/dt to dx/dt. This gives the gradient of the tangent at any point where the parameter is known.

要求参数曲线的斜率，我们不能直接对 y 关于 x 求导。相反，我们使用链式法则将 dy/dx 写成 dy/dt 与 dx/dt 的比值。这样可以在已知参数处求出切线的斜率。

dy/dx = (dy/dt) ÷ (dx/dt) = (dy/dt) / (dx/dt)

This formula is valid provided that dx/dt ≠ 0. If dx/dt = 0 at a particular t-value, the tangent is vertical or the curve has a cusp or singular point, and the gradient is undefined in the usual sense.

该公式在 dx/dt ≠ 0 时成立。如果 dx/dt 在某个 t 值处等于 0，则切线为垂直或曲线有尖点或奇点，通常意义下斜率没有定义。

4. Second Derivative: Concavity and Acceleration | 二阶导数：凹凸性与变化率

For concavity and the classification of stationary points, we often need the second derivative d²y/dx². With parametric equations, this is found by differentiating dy/dx with respect to t and then dividing by dx/dt. The result is still a function of t.

为了判断凹凸性和区分驻点类型，我们通常需要二阶导数 d²y/dx²。对于参数方程，可以通过先对 dy/dx 关于 t 求导，再除以 dx/dt 来得到。结果仍然是 t 的函数。

d²y/dx² = [d/dt(dy/dx)] / (dx/dt)

A very common exam mistake is to compute d²y/dt² and divide it by d²x/dt². This is not correct, because the second derivative is not the ratio of second derivatives. You must differentiate the first derivative dy/dx with respect to t, then divide by dx/dt.

一个非常常见的考试错误是计算 d²y/dt² 并除以 d²x/dt²。这是不正确的，因为二阶导数并不是二阶导数的比。你必须先对一阶导数 dy/dx 关于 t 求导，再除以 dx/dt。

5. Tangents and Normals | 切线与法线

Once dy/dx is known as a function of t, the tangent at a point with parameter t has gradient equal to dy/dx evaluated at that t. The coordinates of the point are simply x = f(t) and y = g(t).

一旦 dy/dx 表示为 t 的函数，参数为 t 的点处切线斜率就等于该 t 值下 dy/dx 的值。该点坐标就是 x = f(t) 和 y = g(t)。

To find the tangent equation, substitute the gradient and coordinates into the straight line formula y – y₁ = m(x – x₁). For the normal, use the negative reciprocal of the tangent gradient, so m_normal = -1 / (dy/dx), provided the tangent gradient is not zero.

求切线方程时，将斜率和坐标代入直线公式 y – y₁ = m(x – x₁)。对于法线，使用切线斜率的负倒数，即 m_normal = -1 / (dy/dx)，前提是切线斜率不等于零。

Edexcel marking schemes often require the final answer to be given in a specific form, such as ax + by + c = 0 or y = mx + c. Always read the question carefully and give the equation in the requested form.

Edexcel 评分标准通常要求最终答案以特定形式给出，例如 ax + by + c = 0 或 y = mx + c。务必仔细审题，并按题目要求的形式给出直线方程。

6. Stationary Points | 驻点

A stationary point occurs where the gradient of the curve is zero, so dy/dx = 0. For parametric curves, this usually means dy/dt = 0, provided dx/dt ≠ 0 at the same t-value. Solving dy/dt = 0 gives the parameter values for any stationary points.

驻点出现在曲线斜率为零处，即 dy/dx = 0。对于参数曲线，这通常意味着 dy/dt = 0，同时要求在该 t 值处 dx/dt ≠ 0。解 dy/dt = 0 可以得到驻点对应的参数值。

To classify a stationary point, you can examine the sign of dy/dx on either side of the point, or use the second derivative test if d²y/dx² is easy to find. A positive second derivative indicates a local minimum, while a negative second derivative indicates a local maximum.

判断驻点是极大还是极小，可以考察该点两侧 dy/dx 的符号，或者在二阶导数易于求得时使用二阶导数判别法。二阶导数为正表示局部极小，二阶导数为负表示局部极大。

Edexcel examiners accept a sign-change table for classification, which is often clearer and avoids the need to compute a complicated second derivative.

Edexcel 考官接受用符号变化表来判断驻点类型，这通常更清晰，也能避免计算复杂的二阶导数。

7. Integration: Area Under a Parametric Curve | 积分：参数曲线下面积

To find the area between a parametric curve and the x-axis, we start with the standard area integral ∫ y dx and convert it into an integral with respect to t. Since x = f(t), we have dx = (dx/dt) dt, so the area becomes an integral involving y and dx/dt.

求参数曲线与 x 轴之间的面积，我们从标准面积积分 ∫ y dx 出发，将其转换为对 t 的积分。由于 x = f(t)，有 dx = (dx/dt) dt，因此面积变为包含 y 和 dx/dt 的积分。

Area = ∫ y dx = ∫ y (dx/dt) dt, from t = t₁ to t = t₂

The limits t₁ and t₂ are the parameter values that correspond to the x-values at the ends of the required interval. You must convert the x-limits to t-limits before integrating, otherwise the integral will not be correct.

积分限 t₁ 和 t₂ 是对应于所求区间端点 x 值的参数值。在积分之前必须将 x 积分限转换为 t 积分限，否则积分将不正确。

If the curve crosses the x-axis within the interval, the signed area may cancel. To find the total actual area, split the integral into sections above and below the x-axis, or use the absolute value of the integral in each section.

如果曲线在区间内穿过 x 轴，有向面积可能会相互抵消。要求总实际面积，应将积分分段处理 x 轴上方和下方的部分，或在各段中使用积分的绝对值。

8. Direction and Limits | 方向与积分限

When x increases from a to b, the parameter t does not always increase. It may decrease depending on the parametric definition of the curve. Therefore, you must not assume that the lower t-limit is smaller than the upper t-limit.

当 x 从 a 增大到 b 时，参数 t 并不总是增大。它可能根据曲线的参数定义而减小。因此，不应假设 t 的下限一定小于上限。

If t decreases

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Parametric Equations: Differentiation and Integration | 参数方程：求导与积分