Parametric Curves & Calculus

Parametric Curves & Calculus

Part 1 of 7 — Parametric Equations

What Are Parametric Equations?

Instead of $y = f(x)$, we express both coordinates in terms of a parameter $t$:

$x = f(t), quad y = g(t)$

First Derivative

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

Worked Example

$x = t^2$, $y = t^3$

$\frac{dx}{dt} = 2t$, $\frac{dy}{dt} = 3t^2$

$\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$

At $t = 2$: slope $= 3$.

Parametric Derivatives 🎯

Key Takeaways — Part 1

1. $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

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 $t$ of $\frac{dy}{dx}$, then divide by .

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

$x = 3cos t$, $y = 3sin t$, $0leqtl$

Parametric Curves

Part 4 of 7 — Area Under Parametric Curves

Area Formula

$A = int_a^b y(t),\frac{dx}{dt},dt$

Parametric Curves

Part 5 of 7 — Eliminating the Parameter

Converting to Rectangular

Sometimes useful for understanding the shape:

Parametric Curves

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Parametric Curves — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Parametric Curves — Complete! ✅