Step By Step Calculus » 12.3 - Differentiation and Cauchy Mean Value Theorem

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Synopsis
For the curves represented by parametric equations x=x(t), y=y(t),x=x(t), y=y(t),
\displaystyle \frac{dy}{dx}=\frac{y'(t)}{x'(t)}\displaystyle \frac{dy}{dx}=\frac{y'(t)}{x'(t)}       provided both x'(t), y'(t)x'(t), y'(t) exist and x'(t)\neq 0x'(t)\neq 0\displaystyle \qquad \displaystyle \qquad
(1)
and similarly
\displaystyle \frac{dx}{dy}=\frac{x'(t)}{y'(t)}\displaystyle \frac{dx}{dy}=\frac{x'(t)}{y'(t)}       provided both x'(t), y'(t)x'(t), y'(t) exist and y'(t)\neq 0y'(t)\neq 0.\displaystyle \displaystyle
From the above definition of first derivative for a parametric curve, one can infer that
  • A horizontal tangent to a parametric curve exists at points where either (i) \displaystyle \frac{dy}{dt}=0\displaystyle \frac{dy}{dt}=0 but \displaystyle \frac{dx}{dt}\neq 0\displaystyle \frac{dx}{dt}\neq 0 or (ii) \displaystyle \frac{dy}{dt}\neq \pm \infty\displaystyle \frac{dy}{dt}\neq \pm \infty but \displaystyle \frac{dx}{dt}=\pm \infty\displaystyle \frac{dx}{dt}=\pm \infty or (iii) \displaystyle \frac{dy}{dx}=0\displaystyle \frac{dy}{dx}=0 when \displaystyle \frac{dy/dt}{dx/dt}\displaystyle \frac{dy/dt}{dx/dt} takes \dfrac{0}{0}\dfrac{0}{0} or \dfrac{\infty}{\infty}\dfrac{\infty}{\infty} form.
  • A vertical tangent to a parametric curve exists at points where either (i) \displaystyle \frac{dx}{dt}=0\displaystyle \frac{dx}{dt}=0 but \displaystyle \frac{dy}{dt}\neq 0\displaystyle \frac{dy}{dt}\neq 0 or (ii) \displaystyle \frac{dx}{dt}\neq\pm\infty\displaystyle \frac{dx}{dt}\neq\pm\infty but \displaystyle \frac{dy}{dt}=\pm\infty\displaystyle \frac{dy}{dt}=\pm\infty or (iii) \displaystyle \frac{dx}{dy}=0\displaystyle \frac{dx}{dy}=0 when \displaystyle \frac{dx/dt}{dy/dt}\displaystyle \frac{dx/dt}{dy/dt} takes \dfrac{0}{0}\dfrac{0}{0} or \dfrac{\infty}{\infty}\dfrac{\infty}{\infty} form.
One can compute the second derivative \displaystyle \frac{d^2y}{dx^2}\displaystyle \frac{d^2y}{dx^2} of a parametric curve using
\displaystyle \frac{d^2y}{dx^2}=\frac{\displaystyle \frac{d}{dt}\left(\frac{y'(t)}{x'(t)}\right)}{x'(t)}.
\displaystyle \frac{d^2y}{dx^2}=\frac{\displaystyle \frac{d}{dt}\left(\frac{y'(t)}{x'(t)}\right)}{x'(t)}.
A curve is concave up in a region if \frac{d^2y}{dx^2}>0\frac{d^2y}{dx^2}>0 in that region, and is concave down in a region if \frac{d^2y}{dx^2}<0\frac{d^2y}{dx^2}<0 in that region. In summary,
\[ \newcommand{\T}{\rule{0pt}{2.6ex}} \newcommand{\B}{\rule[-1.8ex]{0pt}{0pt}} \begin{array}{c|c} \textrm{Condition} & \textrm{Concavity} \\\hline \T\B\frac{d^2y}{dx^2}>0 & \smile \textrm{ concave up}\\ \hline \T\B\frac{d^2y}{dx^2}<0 & \frown \textrm{ concave down}\\ \hline \end{array} \]
A curve is concave right in a region if \frac{d^2x}{dy^2}>0\frac{d^2x}{dy^2}>0 in that region, and is concave left in a region if \frac{d^2x}{dy^2}<0\frac{d^2x}{dy^2}<0 in that region.
\[ \newcommand{\T}{\rule{0pt}{2.6ex}} \newcommand{\B}{\rule[-1.8ex]{0pt}{0pt}} \begin{array}{c|c} \textrm{Condition} & \textrm{Concavity} \\\hline \T\B\frac{d^2x}{dy^2}>0 & ( \;\;\textrm{ concave right}\\ \hline \T\B\frac{d^2x}{dy^2}<0 & ) \;\;\textrm{ concave left}\\ \hline \end{array} \]