Unit 9: Parametric, Polar & Vector-Valued Functions (BC)

9.1 Differentiating parametric equations (BC)

When a curve is traced by x(t) and y(t), the slope is still rise over run — but both rise and run are measured against the parameter t.

9.2 Second derivatives of parametric equations (BC)

To get \dfrac{d^2y}{dx^2} you differentiate the SLOPE \dfrac{dy}{dx} again with respect to x — but since everything is in terms of t, differentiate with respect to t and divide by \dfrac{dx}{dt} once more.

9.3 Arc length of a parametric curve (BC)

The length of a parametric path adds up tiny hypotenuses: each has horizontal piece x'(t)\,dt and vertical piece y'(t)\,dt, so its length is \sqrt{x'^2+y'^2}\,dt.

9.4 Differentiating vector-valued functions (BC)

A vector-valued function \langle x(t),y(t)\rangle traces a moving point.

9.5 Integrating vector-valued functions (BC)

Integrating a velocity vector recovers position, one component at a time.

9.6 Motion with parametric/vector functions (BC)

Planar motion bundles two 1-D motions.

9.7 Polar coordinates and differentiation (BC)

A polar curve r=f(\theta) is really a parametric curve: x=r\cos\theta,\ y=r\sin\theta.

9.8 Area inside a single polar curve (BC)

Sweeping a ray from the origin, each thin slice is nearly a circular sector of radius r(\theta) and tiny angle d\theta.

9.9 Area between two polar curves (BC)

For the region between an outer and inner polar curve, each sector is a washer: subtract the inner sector area from the outer over the angles where the region lives.