11.2 SUMMARY

• A parametric curve \(c(t) = (f(t), g(t))\) describes the path of a particle moving along a curve as a function of the parameter \(t\).
• Parametrizations are not unique: Every curve \(C\) can be parametrized in infinitely many ways. Furthermore, the path \(c(t)\) may traverse all or part of \(C\) more than once.
• Slope of the tangent line at \(c(t)\): \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)} \qquad\quad \textrm{(valid if \(x'(t)\ne 0\))} \]
• Do not confuse the slope of the tangent line \(dy/dx\) with the derivatives \(dy/dt\) and \(dx/dt\), with respect to \(t\).
• Standard parametrizations:
• – Line of slope \(m=s/r\) through \(P=(a,b)\): \(c(t) = (a+rt,b+st)\).
• – Circle of radius \(R\) centered at \(P=(a,b)\): \(c(t) = (a+R\cos t, b+R\sin t)\).
• – Cycloid generated by a circle of radius \(R\): \(c(t) = (R(t-\sin t), R(1-\cos t))\).