Parametric Equations and Parametric Differentiation | 参数方程与参数微分法

📚 Parametric Equations and Parametric Differentiation | 参数方程与参数微分法

Parametric equations are a central topic in Edexcel A Level Mathematics. They allow us to describe curves in the x-y plane using a third variable, usually t. In the exam, you will be expected to convert between parametric and Cartesian forms, differentiate parametrically, find tangents and normals, and use integration to calculate areas under parametric curves.

参数方程是爱德思 A Level 数学的核心内容之一。它利用第三个变量（通常为 t）描述 x-y 平面内的曲线。考试中你需要掌握参数方程与直角坐标方程之间的转换、参数微分法、求切线和法线，以及用积分计算参数曲线下的面积。

1. What Are Parametric Equations? | 什么是参数方程？

A parametric curve is defined by two equations x = f(t) and y = g(t), where t is called the parameter. As t varies, the point (x, y) traces out a curve. This approach is especially useful for curves that are not functions of x, such as circles, ellipses, and many particle motion paths.

参数曲线由两个方程 x = f(t) 和 y = g(t) 定义，其中 t 称为参数。当 t 变化时，点 (x, y) 就描绘出一条曲线。这种方法对于不是 x 的函数的曲线特别有用，例如圆、椭圆以及许多质点运动轨迹。

For example, the unit circle can be written as x = cos t and y = sin t with 0 ≤ t < 2π. This is much neater than the Cartesian form x² + y² = 1 if we need to describe motion along the circle.

例如，单位圆可以写成 x = cos t 和 y = sin t，其中 0 ≤ t < 2π。如果我们需要描述沿圆的运动，这比直角坐标形式 x² + y² = 1 更简洁。

In Edexcel questions, t is often restricted to an interval such as t ∈ [0, 2π] or t ∈ R. Always check the given range, because it determines the portion of the curve you are working with.

在爱德思考题中，t 通常被限制在某个区间，例如 t ∈ [0, 2π] 或 t ∈ R。一定要检查给定的范围，因为它决定了你所研究的曲线部分。

2. Converting to Cartesian Form | 化为直角坐标方程

To convert parametric equations into Cartesian form, eliminate the parameter t. Common methods include substitution, using trigonometric identities, or solving for t in one equation and substituting into the other.

要把参数方程化为直角坐标方程，需要消去参数 t。常用方法包括代入法、使用三角恒等式，或者从一个方程中解出 t 再代入另一个方程。

If x = t + 1 and y = 2t², then t = x – 1, so y = 2(x – 1)². This gives the Cartesian equation directly.

如果 x = t + 1 且 y = 2t²，那么 t = x – 1，因此 y = 2(x – 1)²。这样就可以直接得到直角坐标方程。

For trigonometric parametric equations, use identities such as sin² t + cos² t = 1. If x = 3 cos t and y = 3 sin t, then x² + y² = 9 cos² t + 9 sin² t = 9, giving a circle of radius 3.

对于三角参数方程，使用 sin² t + cos² t = 1 等恒等式。如果 x = 3 cos t 且 y = 3 sin t，则 x² + y² = 9 cos² t + 9 sin² t = 9，得到半径为 3 的圆。

Sometimes a Cartesian equation cannot be expressed as a single neat function, or the question may only ask for a simplified form involving both x and y. Always look for the most direct elimination route.

有时直角坐标方程无法表示为一个简洁的函数，或者题目只要求一个包含 x 和 y 的简化形式。始终寻找最直接的消元途径。

3. Domain and Range from Parameters | 参数下的定义域与值域

The domain and range of a parametric curve are determined by the possible values of x and y as t runs through its given interval. Do not assume the Cartesian equation has the same natural domain.

参数曲线的定义域和值域由 t 在给定区间内变化时 x 和 y 的可能取值决定。不要假设直角坐标方程的自然定义域与之相同。

For example, x = t² and y = t + 2 with t ∈ [-1, 2] gives x ∈ [0, 4] but not all real x. The Cartesian equation y = ±√x + 2 would have a wider domain if taken alone, so the parameter range is essential.

例如，x = t² 和 y = t + 2，其中 t ∈ [-1, 2]，得到 x ∈ [0, 4]，但并非所有实数 x。如果单独看直角坐标方程 y = ±√x + 2，其定义域会比实际更宽，因此参数范围至关重要。

In exam questions, you may be asked to state the domain of the Cartesian form. You must link the range of t to the corresponding range of x or y.

在考题中，你可能会被要求写出直角坐标形式的定义域。你必须把 t 的范围与对应的 x 或 y 的范围联系起来。

t interval	x = t²	y = t + 2
[-1, 2]	[0, 4]	[1, 4]

Always sketch or tabulate key values of t to see the exact portion of the curve being described.

始终画出或列出 t 的关键取值，以看清所描述曲线的确切部分。

4. Parametric Differentiation: First Derivative | 参数微分法：一阶导数

If x = f(t) and y = g(t) are differentiable, then the gradient dy/dx is given by the chain rule:

如果 x = f(t) 和 y = g(t) 可导，则由链式法则得到梯度 dy/dx：

dy/dx = (dy/dt) ÷ (dx/dt)

This formula is valid provided dx/dt ≠ 0. It is often the most efficient way to find the gradient at a point on a parametric curve.

这个公式在 dx/dt ≠ 0 时成立。它通常是求参数曲线上某点梯度最有效的方法。

For x = t² + 1 and y = t³ – 2t, we have dx/dt = 2t and dy/dt = 3t² – 2. Therefore dy/dx = (3t² – 2) / (2t).

对于 x = t² + 1 和 y = t³ – 2t，我们有 dx/dt = 2t，dy/dt = 3t² – 2。因此 dy/dx = (3t² – 2) / (2t)。

Never forget to differentiate with respect to t first, then divide. Also be careful with the condition dx/dt = 0, because the tangent may be vertical there.

切勿忘记先对 t 求导，然后再相除。还要注意 dx/dt = 0 的情况，因为此时切线可能是竖直的。

You can then evaluate dy/dx at a specific t value, for example at t = 1, dy/dx = (3 – 2) / 2 = 1/2.

然后你可以在特定的 t 值处求出 dy/dx，例如在 t = 1 处，dy/dx = (3 – 2) / 2 = 1/2。

5. Finding the Second Derivative d²y/dx² | 求二阶导数 d²y/dx²

The second derivative measures the rate of change of the gradient and helps identify concavity and maximum or minimum points. It is not simply d²y/dt² divided by d²x/dt².

二阶导数衡量梯度的变化率，有助于判断凹凸性和极大或极小点。它并不是简单地用 d²y/dt² 除以 d²x/dt²。

The correct formula is:

正确的公式是：

d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] ÷ (dx/dt)

That is, differentiate the expression for dy/dx with respect to t, then divide by dx/dt. This follows from the chain rule.

也就是说，先对 dy/dx 的表达式关于 t 求导，再除以 dx/dt。这由链式法则得出。

For x = t² + 1 and y = t³ – 2t, dy/dx = (3t² – 2) / (2t). Differentiate this with respect to t using the quotient rule:

对于 x = t² + 1 和 y = t³ – 2t，dy/dx = (3t² – 2) / (2t)。使用商法则对其关于 t 求导：

d/dt [(3t² – 2) / (2t)] = [12t² – (3t² – 2)(2)] / (4t²) = [12t² – 6t² + 4] / (4t²) = (6t² + 4) / (4t²) = (3t² + 2) / (2t²)

Then d²y/dx² = [(3t² + 2) / (2t²)] ÷ (2t) = (3t² + 2) / (4t³).

然后 d²y/dx² = [(3t² + 2) / (2t²)] ÷ (2t) = (3t² + 2) / (4t³)。

A common mistake is to omit the division by dx/dt, so always include this step in your working.

一个常见错误是遗漏除以 dx/dt 的步骤，因此在计算过程中一定要包括这一步骤。

6. Tangents and Normals to Parametric Curves | 参数曲线的切线与法线

To find the tangent or normal at a point on a parametric curve, first identify the value of t at that point, then calculate dy/dx at that t value.

要找到参数曲线上某点的切线或法线，首先确定该点对应的 t 值，然后计算该 t 值处的 dy/dx。

The gradient of the tangent is m = dy/dx. The gradient of the normal is -1/m, provided m ≠ 0. Use the point-slope form y – y₁ = m(x – x₁).

切线斜率为 m = dy/dx。法线斜率为 -1/m，前提是 m ≠ 0。使用点斜式 y – y₁ = m(x – x₁)。

If x = t² + 1, y = t³ – 2t, and t = 1, the point is (2, -1). We found dy/dx = 1/2. The tangent equation is y + 1 = (1/2)(x – 2), which simplifies to y = x/2 – 2.

如果 x = t² + 1，y = t³ – 2t，且 t = 1，点为 (2, -1)。我们已得 dy/dx = 1/2。切线方程为 y + 1 = (1/2)(x – 2)，化简为 y = x/2 – 2。

The normal gradient is -2, so the normal equation is y + 1 = -2(x – 2), or y = -2x + 3.

法线斜率为 -2，因此法线方程为 y + 1 = -2(x – 2)，即 y = -2x + 3。

Sometimes a tangent is vertical. This occurs when dx/dt = 0 and dy/dt ≠ 0. In that case the tangent equation is x = x₁.

有时切线是竖直的。这发生在 dx/dt = 0 且 dy/dt ≠ 0 时。此时切线方程为 x = x₁。

7. Areas Under Parametric Curves | 参数曲线下的面积

To find the area under a parametric curve, use the formula:

要求参数曲线下的面积，使用以下公式：

Area = ∫ y dx = ∫ y(t) · (dx/dt) dt

The limits of integration must be t values corresponding to the x limits. If x = f(t) is increasing, integrate from t₁ to t₂ where x(t₁) = a and x(t₂) = b.

积分限必须是与 x 限相对应的 t 值。如果 x = f(t) 单调递增，则从 t₁ 到 t₂ 积分，其中 x(t₁) = a，x(t₂) = b。

For example, the area under the curve x = t², y = 2t for t from 0 to 1 can be found as follows. Since dx/dt = 2t, the area is ∫₀¹ (2t)(2t) dt = ∫₀¹ 4t² dt = [4t³/3]₀¹ = 4/3.

例如，曲线 x = t²，y = 2t 在 t 从 0 到 1 下的面积可以如下求出。由于 dx/dt = 2t，面积为 ∫₀¹ (2t)(2t) dt = ∫₀¹ 4t² dt = [4t³/3]₀¹ = 4/3。

Always convert the x limits into t limits before integrating. A common error is using x values as t limits directly.

在积分前一定要将 x 限转换为 t 限。一个常见错误是直接把 x 值当作 t 限使用。

If the curve goes below the x-axis, the integral may be negative. Take absolute values or split the region to represent the true geometric area.

如果曲线位于 x 轴下方，积分可能为负。取绝对值或分割区域以表示真实的几何面积。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

Here are the most common errors students make in parametric questions and how to avoid them:

以下是学生在参数方程题中最常见的错误以及如何避免它们：

Forgetting to divide by dx/dt when finding dy/dx.

求 dy/dx 时忘记除以 dx/dt。
Using the wrong formula for d²y/dx², such as (d²y/dt²) ÷ (d²x/dt²).

对 d²y/dx² 使用了错误公式，例如 (d²y/dt²) ÷ (d²x/dt²)。
Not converting x limits to t limits in the area formula.

在面积公式中未将 x 限转换为 t 限。
Ignoring the given range of t and sketching a curve beyond the valid portion.

忽略给定的 t 范围，画出了超出有效部分的曲线。
Losing marks by not simplifying fractions or by using rounded decimals instead of exact values.

因为未化简分数或使用四舍五入的小数而非精确值而失分。

In the exam, show all your differentiation steps clearly. Even if your final answer is wrong, method marks are available for correct derivatives and correct substitution.

考试中要清晰地展示所有求导步骤。即使最终答案错误，正确的导数表达式和正确的代入也能获得方法分。

Also check your tangent point: if t is given, compute x and y before writing the line equation. Many students lose easy marks by using t as the coordinate.

还要检查切点：如果给出了 t，先计算 x 和 y，再写出直线方程。许多学生因把 t 当作坐标而丢分。

9. Worked Example: Mixing Key Skills | 综合例题：混合关键技能

A curve is defined by x = t² + 2t and y = t³ – 3t. Find the Cartesian equation, the gradient at t = 1, and the equation of the tangent at the point where t = 1.

曲线由 x = t² + 2t 和 y = t³ – 3t 定义。求直角坐标方程、t = 1 处的梯度，以及 t = 1 对应点处的切线方程。

First, find the Cartesian equation. From x = t² + 2t, rewrite as x + 1 = (t + 1)², so t = -1 ± √(x + 1). Substituting into y is possible but not needed for the gradient question. For the gradient, use parametric differentiation.

首先求直角坐标方程。由 x = t² + 2t，改写为 x + 1 = (t + 1)²，所以 t = -1 ± √(x + 1)。代入 y 是可行的，但对于梯度题并非必需。对于梯度，使用参数微分法。

We have dx/dt = 2t + 2 and dy/dt = 3t² – 3. Therefore:

我们有 dx/dt = 2t + 2 和 dy/dt = 3t² – 3。因此：

dy/dx = (3t² – 3) / (2t + 2) = 3(t² – 1) / [2(t + 1)] = 3(t – 1)(t + 1) / [2(t + 1)] = 3(t – 1) / 2, for t ≠ -1

At t = 1, dy/dx = 3(0) / 2 = 0. The tangent is horizontal at this point.

在 t = 1 处，dy/dx = 3(0) / 2 = 0。切线在该点处是水平的。

The point when t = 1 is x = 1² + 2(1) = 3 and y = 1³ – 3(1) = -2, so the point is (3, -2). The tangent equation is y = -2.

当 t = 1 时，x = 1² + 2(1) = 3，y = 1³ – 3(1) = -2，因此点为 (3, -2)。切线方程为 y = -2。

This example shows that simplifying the gradient expression before substitution can make the solution much easier.

这个例子说明，在代入之前先化简梯度表达式可以使求解容易得多。

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Parametric Equations and Parametric Differentiation | 参数方程与参数微分法