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\problemgrouptitle{Parametric Equations Review}

\begin{problemblock}{Find the equation of the tangent line to the curve at the point corresponding to the given value of the parameter.}

\item $x=2t+e^{-t},\ \ \ y=3 e^t;\ \ \ \ t=0$
\answer{
$y=3x-2$
}

\item $x=t+\sin t,\ \ \ y=\cos t;\ \ \ \ t=\frac{\pi}{2}$
\answer{
$\displaystyle y=-x+\frac{\pi}{2}+1$
}

\end{problemblock}

\begin{problemblock}{Find the points on the curve at which the tangent
line is horizontal or vertical.}

\item $x=e^t-3t,\ \ \ y=e^{-2t}+4t$
\answer{
Horizontal if $t=-\ln 2$,\ \ Vertical if $t=\ln 3$
}

\item $x=t-2\sin t,\ \ \ y=t+\cos t;\,\,\, 0\le t\le 2\pi$
\answer{
Horizontal if $\displaystyle t=\frac{\pi}{2}$,\ \ Vertical if $\displaystyle t=\frac{\pi}{3}$, \,\,\,$\displaystyle t=\frac{5\pi}{3}$
}

\end{problemblock}

\begin{problemblock}{Find the area enclosed by the given curves.}

\item $x=t+5$,\ $y=\ln t$; \  the $x$-axis, and the line $x=15$\,.

\answer{
$A=10 \ln 10-9$
}

\item $x= \frac 12t^2+3$,\ $y=3t+e^{-t}-1$;\  the $x$-axis, and the line $x=11$\,.
\answer{
$A=61-5e^{-4}$
}

\end{problemblock}

\begin{problemblock}{Find the length of the given curve.}

\item $x=\frac{1}{2} t^2-\ln t,\ \ \ y=2t; \,\,\, 1\le t\le 5$
\answer{
$s=12+\ln 5$
}

\item $x= e^t \cos t,\ \ \ y=e^t \sin t; \,\,\, 0\le t\le \pi$
\answer{
$\displaystyle s=\sqrt 2\left(e^{\pi}-1 \right)$
}

\end{problemblock}

\textbook{\addtoanswerkey{\clearpage}}

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