Parametric Equations
Explore parametric equations where x and y are both functions of a parameter t, enabling representation of curves that fail the vertical line test. Calculate derivatives dy/dx = (dy/dt)/(dx/dt) and second derivatives for analyzing concavity. Find arc length, speed, and velocity vectors. Visualize classic parametric curves like cycloids, ellipses, and Lissajous figures, and understand applications in physics for modeling projectile motion and planetary orbits.
Polar Area & Curves
Graph and analyze curves in polar coordinates where r = f(θ), and calculate areas using A = (1/2)∫[α to β] [r(θ)]²dθ. Visualize classic polar curves including circles, cardioids, limaçons, rose curves, and lemniscates. Understand symmetry tests, find tangent lines using dr/dθ, and explore how polar coordinates naturally describe curves with rotational symmetry that are complex in Cartesian form.