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Calculus Small Changes, Big Ideas

Bill Shillito

Section 6.2 Parametric curves

Introduction goes here.
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  • Parametric curves are traced out by the motion of a particle, where \(x\) and \(y\) are given in terms of \(t\)
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  • Eliminating the parameter
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  • Slope of a parametric curve:
    \begin{equation*} \dv{y}{x}=\frac{\dd{y}/\dd{t}}{\dd{x}/\dd{t}} \end{equation*}
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  • Parametrization of ellipse:
    \begin{equation*} \frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \end{equation*}
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    \begin{align*} x \amp = h + a\cos t \\ y \amp = k + b\sin t \end{align*}
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  • Parametrization of hyperbola:
    \begin{equation*} \frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \end{equation*}
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    \begin{align*} x \amp = h + a\sec t \\ y \amp = k + b\tan t \end{align*}
    (Or switch \(\tan\) or \(\sec\) for the other one)
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