10.1 Definition of Parametric Equations

Parameter: $t$ is a parameter in functions $f(t)$ and $g(t)$

Parametric Equations: $x=f(t), y=g(t)$ where $t$ is time

Graph $(x,y)$ values for $t\in[a, b]$

• Example

Question Types

1. Sketch parametric equation (e.g. Sketch $x=t^2-2t, y=t+1,t \in [0,4]$)

10.2 Calculus with Parametric Curves

Differentiation:

• $\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}$
• Notation: $\frac{dx}{dt}=f',\frac{dy}{dt}=g'$
• Vertical Tangent: $\frac{dx}{dt}=0,\frac{dy}{dt}\ne0$
• Horizontal Tangent: $\frac{dy}{dt}=0,\frac{dx}{dt}\ne0$
• $\frac{d^2y}{dx^2}=\frac{d}{dt}(\frac{dy}{dx})/\frac{dx}{dt}$

Area:

• $A=\int_a^b y \, dx$
• $A_{parametric}=\int^\beta_\alpha g(t)f'(t) \, dt$ (when $t \in [\alpha,\beta]$)

Arc Length:

• $L=\int^b_a\sqrt{1+(\frac{dy}{dx})^2} \, dx$
• $L_{parametric}=\int_\alpha^\beta \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} \, dt$ (requires $f',g'$ to be continuous on $[\alpha,\beta]$, curve traversed exactly once)