Parametric Curves & Calculus - Complete Interactive Lesson
Part 1: Parametric Equations
Parametric Curves & Calculus
Part 1 of 7 — Parametric Equations
What Are Parametric Equations?
Instead of , we express both coordinates in terms of a parameter :
First Derivative
Worked Example
,
,
At : slope .
Parametric Derivatives 🎯
Key Takeaways — Part 1
Part 2: Second Derivative
Parametric Curves
Part 2 of 7 — Second Derivative
Second Derivative for Parametric Curves
\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}
Take the derivative with respect to of , then divide by .
Part 3: Arc Length (Parametric)
Parametric Curves
Part 3 of 7 — Arc Length (Parametric)
Parametric Arc Length Formula
L = int_a^b sqrt{left(\frac{dx}{dt}\right)^2 + left(\frac{dy}{dt}\right)^2},dt
Example: Circle
, ,
Part 4: Area Under Parametric Curves
Parametric Curves
Part 4 of 7 — Area Under Parametric Curves
Area Formula
Part 5: Eliminating the Parameter
Parametric Curves
Part 5 of 7 — Eliminating the Parameter
Converting to Rectangular
Sometimes useful for understanding the shape:
| Parametric | Rectangular |
|---|---|
Part 6: Practice Workshop
Parametric Curves
Part 6 of 7 — Practice Workshop
Mixed Practice 🎯
Workshop Complete!
Part 7: Final Assessment
Parametric Curves — Review
Part 7 of 7 — Final Assessment
Final Assessment 🎯