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\title{ESP Workshop, Worksheet \#14}
\date{\today}
\begin{document}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#14}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item For each of the vector-valued functions given below, find the
  limit. Remember, a vector-valued function is of the form
\begin{displaymath}
\mathbf{r}(t)= f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k}
\end{displaymath}
where $f,g,$ and $h$ are real-valued functions, i.e. $f,g,h:\mathbb{R}
\rightarrow \mathbb{R}$.
\begin{enumerate}
\item $\lim _{t\rightarrow 0^+} \langle \cos t, \sin t, t \ln t \rangle$
\item $\lim _{t\rightarrow 0} \langle \frac{e^t -1}{t}, \frac{\sqrt{1+t} -1}{t}, \frac{3}{1+t} \rangle$
\item $\lim _{t\rightarrow 1} \left( \sqrt{t+3} \mathbf{i} + \frac{t-1}{t^2 -1} \mathbf{j} +\frac{\tan t}{t} \mathbf{k} \right)$
\end{enumerate}

\item Match the graphs of the parametric equations $x=f(t)$ and
  $y=g(t)$ in (a)--(d) with the parametric curves labeled I--IV. Give
  reasons for your choices.

\vspace{2.5in}

\item Use the graphs of $x=f(t)$ and $y=g(t)$ to sketch the parametric
  curve $x=f(t),\ y=g(t)$. Indicate with arrows the direction in which
  the curve is traced as $t$ increases.

\newpage

\item A curve, called a \textbf{witch of Maria Agnesi}, consists of
  all possible positions of the point $P$ in the figure. (See also the
  movie...)
\vspace{2.2in}

\begin{enumerate}
\item Show that parametric equations for this curve can be written as
\begin{displaymath}
x = 2a \cot \theta \quad y = 2a \sin^2 \theta
\end{displaymath}

\item Sketch the curve.

\item Recall that Simpson's Rule says that
\begin{displaymath}
\int _a ^b f(x) dx \approx \frac{\Delta x}{3} \left[ f(x_0) +4 f(x_1) +2 f(x_2) + 4f(x_3) +\cdots +2f(x_{n-2}) + 4f(x_{n-1}) +f(x_n) \right] 
\end{displaymath}
where $n$ is even and $\Delta x = (b-a)/n$. Use Simpson's Rule with
$n=4$ to estimate the length of the arc of this curve with $\pi/4 \leq
\theta \leq \pi /2$.
\end{enumerate}

\item A string is wound around a circle and then unwound while being
  held taut. The curve traced by the point $P$ at the end of the
  string is called the \textbf{involute} of the circle. If the circe
  has radius $r$ and center $O$ and the initial position of $P$ is
  $(r,0)$, and if the parameter $\theta$ is chosen as in the figure,
  show that the parametric equations of the involute are
\begin{displaymath}
x=r(\cos \theta + \theta \sin \theta) \quad y = r(\sin \theta - \theta \cos \theta)
\end{displaymath}

\vspace{1.5in}

\item (An Agricultural question!) A cow is tied to a silo with radius
  $r$ by a rope just long enough to reach the opposite side of the
  silo. First draw a picture of the situation, and then find the area
  available for grazing by the cow. (Hint: use the ideas from the
  previous question...)

\end{enumerate}

\end{document}