Parametric Equations

Understanding Parametric Equations

Describing curves using parameter-based functions

Core Concept

Parametric equations describe the coordinates of a point (x,y) in terms of a third variable called a parameter, usually denoted as t.

\[x = f(t)\] \[y = g(t)\]

The parameter t acts as a control variable, determining both x and y coordinates as it varies.

Converting Between Forms

Moving between parametric and rectangular forms

Example: Circle Parameterization

Parametric Form

A circle with radius r can be parameterized as:

\[x = r\cos(t)\] \[y = r\sin(t)\] \[0 \leq t \leq 2\pi\]

Converting to Rectangular

Square and add the equations:

\[\frac{x^2}{r^2} + \frac{y^2}{r^2} = \cos^2(t) + \sin^2(t) = 1\] \[x^2 + y^2 = r^2\]

Derivatives in Parametric Form

Finding slopes and rates of change

The Chain Rule Approach

The derivative dy/dx can be found using the chain rule:

\[\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}\]

This represents the slope of the tangent line at any point on the curve.

Example: Finding Slope

Given Equations

\[x = t^2\] \[y = t^3\]

Find dy/dx

\[\frac{dx}{dt} = 2t\] \[\frac{dy}{dt} = 3t^2\]

\[\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}\]

Real-World Applications

Using parametric equations to model motion

Projectile Motion

Horizontal Motion

\[x = v_0\cos(\theta)t\]

Where v₀ is initial velocity and θ is launch angle

Vertical Motion

\[y = v_0\sin(\theta)t - \frac{1}{2}gt^2\]

Where g is acceleration due to gravity (≈ 9.8 m/s²)

Video Explanation

Practice Problems

Test your understanding of parametric equations.