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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Tangents and Areas">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 21:01:28 1997}
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\section{Tangents and Areas\label{tangents}}

\subsubsection{Tangents}

Some curves defined by parametric equations $x=f\left( t\right) $ and $%
y=g\left( t\right) $ can also be expressed, by eliminating the parameter, in
the form $y=F\left( x\right) $. If you substitute the equations $x=f\left(
t\right) $ and $y=g\left( t\right) $ into the equation $y=F\left( x\right) $%
, you obtain 
\[
g\left( t\right) =F\left( f\left( t\right) \right) 
\]
If $g$, $F$, and $f$ are differentiable, then the \hyperref{Chain Rule}{}{}{%
cal02_05.tex#Chain rule} gives 
\[
g\prime \left( t\right) =F\prime \left( f\left( t\right) \right) f\prime
\left( t\right) =F\prime \left( x\right) f\prime \left( t\right) 
\]
If $f\prime \left( t\right) \neq 0$, you can solve for $F\prime \left(
x\right) $: 
\[
F\prime \left( x\right) =\frac{g\prime \left( t\right) }{f\prime \left(
t\right) }
\]
Since the slope of the tangent to the curve $y=F\left( x\right) $ at $\left(
x,F\left( x\right) \right) $ is $F\prime \left( x\right) $, the equation
above enables us to find tangents to parametric curves without having to
eliminate the parameter. This equation can be rewritten in Leibniz notation
as 
\[
\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\text{\quad if\quad }\frac{%
dx}{dt}\neq 0\label{Eq 2}
\]
The curve has a horizontal tangent when $\frac{dy}{dt}=0$ (and $\frac{dx}{dt}%
\neq 0$), and vertical tangent when $\frac{dx}{dt}=0$ (and $\frac{dy}{dt}%
\neq 0$). The second derivative can be found by replacing $y$ in \hyperref{%
the equation above}{}{}{Eq 2} by $\frac{dy}{dx}$: 
\[
\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left( \frac{dy}{dx}\right) =\frac{\frac{d%
}{dt}\left( \frac{dy}{dx}\right) }{\frac{dx}{dt}}
\]

\subsubsection{Areas\label{areas}}

The area under a curve $y=F\left( x\right) $ from $a$ to $b$ is $%
A=\int_{a}^{b}F\left( x\right) dx$, where $F\left( x\right) \geq 0$. If the
curve is described by the parametric equations $x=f\left( t\right) $ and $%
y=g\left( t\right) $, $\alpha \leq t\leq \beta $, then you can use the 
\hyperref{Substitution Rule for Definite Integrals}{}{}{%
cal05_05.tex#substitution rule for definite integrals} as follows: 
\[
A=\int_{a}^{b}ydx=\int_{\alpha }^{\beta }g\left( t\right) f\prime \left(
t\right) dt\text{ [ or}\int_{\beta }^{\alpha }g\left( t\right) f\prime
\left( t\right) dt\text{ ]} 
\]

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