Parametric curves are differentiated through the parameter
A parametric curve gives x and y separately as functions of a parameter t. Its slope is dy/dx = (dy/dt)/(dx/dt), a ratio of the component rates, with horizontal tangents where dy/dt = 0 and vertical tangents where dx/dt = 0. The second derivative comes from differentiating dy/dx with respect to t and dividing by dx/dt again, and the arc length of the curve is ∫√((dx/dt)² + (dy/dt)²) dt.
Parametricdy/dx = (dy/dt)/(dx/dt)Arc Length
Key Concept 2
Vector-valued functions model motion componentwise
A vector-valued function r(t) = ⟨x(t), y(t)⟩ traces a path in the plane. You differentiate and integrate componentwise, so the velocity is v(t) = r′(t), the acceleration is a(t) = r″(t), and the speed is the magnitude |v(t)| = √((dx/dt)² + (dy/dt)²). Total distance traveled is ∫|v| dt, and a position is recovered by integrating velocity and adding an initial position vector.
Vector-ValuedComponentwiseSpeed = |v|
Key Concept 3
Polar coordinates describe points by radius and angle
Polar coordinates (r, θ) locate a point by its distance r from the origin and angle θ, converting with x = r cos θ, y = r sin θ. You can differentiate a polar curve in the form dy/dx = (dy/dθ)/(dx/dθ), and the area enclosed by a single polar curve is A = ½∫ r² dθ (each slice is a circular sector). The area between two polar curves subtracts the squared radii: ½∫(R² − r²) dθ.
Polar (r, θ)½∫ r² dθBetween Curves
Parametric equations
x = x(t), y = y(t) tracing a curve via a parameter t.