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Parametric Equations — Syllabus Points

1. Parametric Differentiation​

  • 1.1 Find dydx\frac{dy}{dx} from parametric equations x=x(t)x=x(t), y=y(t)y=y(t) using dydx=dy/dtdx/dt\frac{dy}{dx}=\frac{dy/dt}{dx/dt}
  • 1.2 Find d2ydx2\frac{d^2y}{dx^2} by differentiating dydx\frac{dy}{dx} with respect to tt and dividing by dxdt\frac{dx}{dt}
  • 1.3 Apply product and chain rules in parametric form
  • 1.4 Evaluate first and second derivatives at given parameter values

2. Arc Length of Parametric Curves​

  • 2.1 Set up the arc length integral L=∫ab(dxdt)2+(dydt)2 dtL=\int_a^b\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\,dt
  • 2.2 Simplify the integrand by completing the square or trigonometric identities
  • 2.3 Evaluate the integral to find the exact length
  • 2.4 Handle limits correctly with parameter tt (not xx or yy)

3. Applications​

  • 3.1 Find tangents and normals to parametric curves
  • 3.2 Determine stationary points on parametric curves
  • 3.3 Connect parametric arc length to Cartesian arc length