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%  Calculus of Parametric Curves                   %
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\problemgrouptitle{Calculus of Parametric Curves}

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

\item $x=3t^2+4,\ \ \ y=2t+1;\ \ \ \ t=1$
\answer{
$\displaystyle y=\frac{1}{3}x+\frac{16}{3}$
}

\item $x=e^{2t}+3,\ \ \ y=e^{-t}+1;\ \ \ \ t=0$
\answer{
$\displaystyle y=-\frac{1}{2}x+4$
}

\item $x=\sec t,\ \  y=2  \tan t ;\ \ \ t=\frac{\pi}{6}$
\answer{
$\displaystyle y=4x-2 \sqrt 3$
}

\item $x=2\cos t,\ \  y=6 \sin t ;\ \ \ t=\frac{\pi}{3}$
\answer{
$\displaystyle y=-\sqrt{3}x+4 \sqrt{3}$
}

\end{problemblock}

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

\item $x=t^3+4,\ \ \ y=t^3-12t+1$
\answer{
Horizontal if $t=\pm 2$,\ \ Vertical if $t=0$
}

\item $x=e^t-3e^{-t}-4t,\ \ \ y=e^{2t}-8t$
\answer{
Horizontal if $t=\ln 2$,\ \ Vertical if $t=0, \,\, t=\ln 3$
}

\end{problemblock}

\begin{problemblock}{Find $\displaystyle\frac {d^2y}{dx^2}$ and analyze the concavity of the following curves.}

\item $x=\sqrt t,\ \ \ y=t^2-3 \sqrt t; \,\,\,\,\,t\ge0$
\answer{
Concave up for all $t\ge0$
}

\item $x=t+2,\ \ \ y= e^t-t^2;\,\,\,\,\,t\ge0$
\answer{
Concave up for $t>\ln 2$ and concave down for $0\le t<\ln 2$
}

\end{problemblock}

\item Find the area enclosed by the $x$-axis and the curve $x=t^2+5,\ \ \ y=8t-t^2;\,\,\, t\in\mathset{R}$\,.
\answer{
$\displaystyle A=\frac {32}{3}$
}

\item Find the area enclosed by the curve $x=\sin t,\ \ \ y= 1+\cos t;\,\,\, 0\le t\le 2\pi$\,.
\answer{
$A=\pi$
}

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

\item $x= t+\frac{1}{t},\ \ \ y=2\ln t; \,\,\, 1\le t\le 10$
\answer{
$\displaystyle s=\frac{99}{10}$
}

\item $x=t^2+3,\ \ \ y= 2t+1;\,\,\, 0\le t\le 1$
\answer{
$\displaystyle s=\sqrt 2+\ln(1+\sqrt 2)$
}

\end{problemblock}

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