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Parametric Functions

Definition

Definition of Parametric Functions

A function relationship determined by parametric equations $\begin{cases} x = x(t) \\ y = y(t) \end{cases}$ is called a parametric function.

数学语言

The rigorous definition of a parametric function is: for each $t$ in the parameter domain $D$ , there exists a uniquely determined $(x, y) = (x(t), y(t))$ , and both $x = x(t)$ and $y = y(t)$ are functions of $t$ .

Parametric functions describe originally complex curves using two simple functional relationships by introducing parameter t, making them easier to study and compute.

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Document
$x(t)$	Mathematical Symbol	x of t	Function of parameter t for x
$y(t)$	Mathematical Symbol	y of t	Function of parameter t for y
$t$	Mathematical Symbol	t	Parameter variable
$D$	Mathematical Symbol	D	Parameter domain, range of parameter t

Characteristics

Parametric functions have the following characteristics:

Parameter representation: Both independent and dependent variables are expressed through parameter t
Parameter elimination possible: Explicit functions can be obtained through parameter elimination
Geometric meaning: Geometrically represents curves
Directionality: The direction of change of parameter t determines the direction of the curve

Common Parametric Equations

Parametric Equations of a Circle

Parametric equations of a circle with radius r: $\begin{cases} x = r\cos t \\ y = r\sin t \end{cases}, \quad t \in [0, 2\pi]$

Parametric Equations of an Ellipse

Parametric equations of an ellipse with semi-major axis a and semi-minor axis b: $\begin{cases} x = a\cos t \\ y = b\sin t \end{cases}, \quad t \in [0, 2\pi]$

Parametric Equations of a Line

Parametric equations of a line passing through point $(x_0, y_0)$ with direction vector $(a, b)$ : $\begin{cases} x = x_0 + at \\ y = y_0 + bt \end{cases}, \quad t \in \mathbb{R}$

$\mathbb{R}$ (Blackboard bold R): This is a standard symbol in mathematics representing the set of real numbers (Real numbers), the set of all real numbers. Blackboard bold is a special font style used in mathematics to denote sets, distinguishing set symbols from ordinary variables.

Parametric Equations of a Cycloid

Parametric equations of a cycloid: $\begin{cases} x = a(t - \sin t) \\ y = a(1 - \cos t) \end{cases}, \quad t \in \mathbb{R}$

Parameter Elimination Methods

Basic Steps

Solve for parameter t from the parametric equations
Substitute the expression for t into the other equation
Simplify to obtain the explicit functional relationship

Example

For the parametric equations of a circle $\begin{cases} x = r\cos t \\ y = r\sin t \end{cases}$ :

From the first equation: $\cos t = \frac{x}{r}$
From the second equation: $\sin t = \frac{y}{r}$
Using $\sin^2 t + \cos^2 t = 1$ : $(\frac{x}{r})^2 + (\frac{y}{r})^2 = 1$
Simplify to obtain: $x^2 + y^2 = r^2$

Derivatives of Parametric Functions

Basic Formula

If $\begin{cases} x = x(t) \\ y = y(t) \end{cases}$ , then:

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{y'(t)}{x'(t)}$

Second Derivative

$\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{d}{dt}(\frac{y'(t)}{x'(t)}) \cdot \frac{dt}{dx}$

Exercises

Exercise 1

Find the explicit functional relationship for the parametric equations $\begin{cases} x = t^2 \\ y = t^3 \end{cases}$ .

Reference Answer(5 个标签)

parametric functionparameter eliminationpower functionexplicit functiondomain

Problem-solving approach: Obtain the explicit function relationship through parameter elimination.

Detailed steps:

Solve for parameter t from the first equation: $t = \sqrt{x}$ (note $x \geq 0$ )
Substitute the expression for t into the second equation: $y = (\sqrt{x})^3 = x^{3/2}$
Therefore, the explicit function is: $y = x^{3/2}$ , domain $[0, +\infty)$

Answer: $y = x^{3/2}$ , domain $[0, +\infty)$ .

Exercise 2

Find the derivative of the parametric equations $\begin{cases} x = \cos t \\ y = \sin t \end{cases}$ .

Reference Answer(5 个标签)

parametric functionderivativetrigonometric functioncircle parametric equationsimplicit differentiation

Problem-solving approach: Use the derivative formula for parametric functions.

Detailed steps:

Calculate $x'(t)$ and $y'(t)$ : $x'(t) = -\sin t$ $y'(t) = \cos t$
Use the derivative formula: $\frac{dy}{dx} = \frac{y'(t)}{x'(t)} = \frac{\cos t}{-\sin t} = -\cot t$
Since $x = \cos t$ , we have $t = \arccos x$ , substitute to get: $\frac{dy}{dx} = -\cot(\arccos x) = -\frac{x}{\sqrt{1-x^2}}$

Answer: $\frac{dy}{dx} = -\frac{x}{\sqrt{1-x^2}}$ .

Exercise 3

Find the explicit functional relationship for the parametric equations $\begin{cases} x = e^t \\ y = e^{-t} \end{cases}$ .

Reference Answer(5 个标签)

parametric functionparameter eliminationexponential functioninverse proportion functionlogarithmic function

Problem-solving approach: Obtain the explicit function relationship through parameter elimination.

Detailed steps:

Solve for parameter t from the first equation: $t = \ln x$ (note $x > 0$ )
Substitute the expression for t into the second equation: $y = e^{-\ln x} = \frac{1}{x}$
Therefore, the explicit function is: $y = \frac{1}{x}$ , domain $(0, +\infty)$

Answer: $y = \frac{1}{x}$ , domain $(0, +\infty)$ .

Summary

Symbols Used in This Document

Symbol	Type	Pronunciation/Explanation	Meaning in This Document
$x(t)$	Mathematical Symbol	x of t	Function of parameter t for x
$y(t)$	Mathematical Symbol	y of t	Function of parameter t for y
$t$	Mathematical Symbol	t	Parameter variable
$\frac{dy}{dx}$	Mathematical Symbol	dy by dx	Derivative of y with respect to x
$\frac{d^2y}{dx^2}$	Mathematical Symbol	d squared y by dx squared	Second derivative of y with respect to x
$x'(t)$	Mathematical Symbol	x prime of t	Derivative of x with respect to t
$y'(t)$	Mathematical Symbol	y prime of t	Derivative of y with respect to t
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, the set of all real numbers
$[0, 2\pi]$	Mathematical Symbol	Closed interval	Closed interval from 0 to $2\pi$

Chinese-English Glossary

Chinese Term	English Term	Pronunciation	Explanation
参数函数	parametric function	/pærəˈmetrɪk ˈfʌŋkʃən/	Functions expressed through parametric equations
参数方程	parametric equation	/pærəˈmetrɪk ɪˈkweɪʒən/	Functional equations expressed using parameters
消参	parameter elimination	/pəˈræmɪtə ɪˌlɪmɪˈneɪʃən/	Eliminating parameters from parametric equations to obtain explicit functions
显式函数	explicit function	/ɪkˈsplɪsɪt ˈfʌŋkʃən/	Functions that can directly express y in terms of x
方向向量	direction vector	/dɪˈrekʃən ˈvektə/	Vector indicating the direction of a line
轨迹	locus	/ˈləʊkəs/	Path traced by a moving point
极坐标	polar coordinates	/ˈpəʊlə kəʊˈɔːdɪneɪts/	Coordinate system using radius and angle
链式法则	chain rule	/tʃeɪn ruːl/	Rule for differentiating composite functions
二阶导数	second derivative	/ˈsekənd dɪˈrɪvətɪv/	Derivative of the first derivative of a function

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