Advanced Placement Calculus BC including series, parametric equations, and polar functions.
Instead of expressing \(y\) as a function of \(x\), parametric equations describe both \(x\) and \(y\) as separate functions of a parameter, usually \(t\):
\[ x = f(t), \quad y = g(t) \]
This is perfect for modeling motion, curves, and paths that can't be described by a single equation.
Slope: The slope at any point is \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\).
Arc Length: Find the length of a curve using
\[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}dt \]
Area: Calculate areas under parametric curves using integrals.
They're used in physics (projectile motion), animation, robotics, and more!
\[\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\]
Projectile motion: \(x = v_0 t \cos \theta\), \(y = v_0 t \sin \theta - \frac{1}{2}gt^2\).
The circle: \(x = r \cos t\), \(y = r \sin t\), for \(0 \leq t \leq 2\pi\).
Parametric equations let us describe complex curves and their behavior using calculus.