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\title{Calculus and Elements of Linear Algebra I}
\author{Homework 6}
\date{Due on Moodle, Monday, October 19, 2020}
\maketitle


\noindent In Problems 1--3, your answer should include the following points:
\begin{enumerate}[label={\upshape(\roman*)}]
\item The domain of the function,
\item $y$-intercept if any, $x$-intercepts when they are easy to compute,
\item horizontal asymptotes,
\item vertical asymptotes,
\item analysis of the first derivative (intervals where the function
is increasing or decreasing, local minima or maxima)
\item analysis of the second derivative (intervals where the function
is concave up or concave down, points of inflection).
\end{enumerate}
Finally, sketch the function.  Your drawing does not need to be to
scale, but should show all the qualitative features of the graph.

\begin{enumerate}

\item Sketch the graph of the function
\[
  f(x) = \frac{x^2}{x^2-1} \,.
\]

\item Sketch the graph of the function
\[
  f(x) = \sqrt x - \ln x \,.
\]

\item Sketch the graph of the function
\[
  f(x) = \mathrm{e}^{-1/x} \,.
\]


\item A farmer owns an area of land adjacent to a lake with a straight
shoreline.  She has $500\,$m of fencing with which to enclose her
cattle.  Assuming that she uses the shore as one side of a rectangular
enclosure, what dimensions should the rectangle have in order that the
cattle have the largest possible area in which to graze?

\item Find the radius of the circle which minimizes the square of the
distance of the circle centered at the origin to the points $(0,1)$
and $(-1,2)$.

\end{enumerate}

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