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Parametric Speed & Arc Length
AP Calculus BC · Unit 9 Parametric Equations
Learning Objectives
Graph parametric curves x(t), y(t)
Calculate position, velocity, and speed
Evaluate arc length distance traveled
Distinguish between path length and displacement
Formulas
Velocity Vector:
v(t) = ⟨x'(t), y'(t)⟩
Speed (Magnitude of v):
|v(t)| = √[x'(t)² + y'(t)²]
Arc Length (Distance):
L = ∫ √[x'(t)² + y'(t)²] dt
Tags
parametric
arc length
speed
particle motion
PARAMETRIC EQUATION
Cycloid: x=t−sin(t), y=1−cos(t)
Ellipse: x=4cos(t), y=2sin(t)
Lissajous: x=sin(3t), y=sin(2t)
Butterfly: x=sin(t)(e^(cos t)−2cos(4t))
Time t
0.00
▶ Animate Particle
Position ⟨x, y⟩
—
Velocity ⟨x', y'⟩
—
Inst. Speed |v(t)|
—
Arc Length (dist)
—
Blue arrow
represents the velocity vector.
Green path
shows arc length accumulated from t=0.