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AP CALCULUS BC · MODULE 05

Parametric, Polar & Vector Functions
Curves Beyond y = f(x)

Learn to analyze curves defined parametrically, in polar coordinates, and as vector-valued functions — essential for BC Calculus.

📘 BUders Learning System
This module covers parametric equations, polar coordinates, and vector-valued functions — including derivatives, arc length, and area calculations. All topics are BC Calculus only.
📐 01
Parametric Equations
$x = f(t)$, $y = g(t)$ — representing curves with a parameter $t$, eliminating the parameter, and sketching.
🔷 BC only → start
⚡ 02
Derivatives of Parametric Curves
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, tangent lines, horizontal and vertical tangents.
🔷 BC only → start
📏 03
Arc Length & Speed (Parametric)
$L = \int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} dt$, speed = $\sqrt{(dx/dt)^2 + (dy/dt)^2}$.
🔷 BC only → start
🎯 04
Polar Coordinates
$(r, \theta)$ — converting between polar and rectangular, sketching polar curves.
🔷 BC only → start
📈 05
Derivatives & Area in Polar Coordinates
$\frac{dy}{dx} = \frac{r' \sin\theta + r\cos\theta}{r' \cos\theta - r\sin\theta}$, $A = \frac{1}{2} \int_\alpha^\beta r^2 d\theta$.
🔷 BC only → start
➡️ 06
Vector-Valued Functions
$\mathbf{r}(t) = \langle x(t), y(t) \rangle$ — derivatives, velocity, acceleration, and motion in the plane.
🔷 BC only → start
📝 07
Practice Problems
20 solved problems covering parametric, polar, and vector functions.
practice → start
🧪 08
Test Yourself
20-question interactive quiz with instant feedback.
assessment → start