📚 Parametric Equations and Differentiation | 参数方程与求导
Parametric equations describe curves by expressing both x and y in terms of a third variable, usually t. In Edexcel A-Level Mathematics, this topic connects algebra, graphs, calculus and coordinate geometry, making it a favourite area for examiners.
参数方程通过用第三个变量（通常是 t）同时表示 x 和 y 来描述曲线。在 Edexcel A-Level 数学中，这一主题将代数、图像、微积分和坐标几何串联起来，因此备受考官青睐。
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, the point (x, y) traces out a curve in the xy-plane.
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 变化时，点 (x, y) 在 xy 平面上勾勒出一条曲线。
For example, x = t² and y = 2t represent the parabola y² = 4x after eliminating t.
例如，x = t² 和 y = 2t 在消去 t 后表示抛物线 y² = 4x。
Parametric representations are useful when a curve is not a single-valued function, or when modelling motion where t represents time.
当曲线不是单值函数，或者用 t 表示时间进行运动建模时，参数表示非常有用。
2. Converting Between Parametric and Cartesian Forms | 参数式与直角坐标式的互化
To find the Cartesian equation, eliminate the parameter t from the two equations. Common strategies include substitution and using trigonometric identities such as sin²θ + cos²θ = 1.
要找到直角坐标方程，需从两个方程中消去参数 t。常用方法包括代入法以及使用三角恒等式，例如 sin²θ + cos²θ = 1。
- If x = t + 1 and y = 2t – 3, write t = x – 1, then substitute into y: y = 2(x – 1) – 3 = 2x – 5. – 若 x = t + 1 且 y = 2t – 3，则写出 t = x – 1，再代入 y：y = 2(x – 1) – 3 = 2x – 5。
- If x = 3cosθ and y = 3sinθ, use cos²θ + sin²θ = 1 to get x² + y² = 9. – 若 x = 3cosθ 且 y = 3sinθ，利用
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导