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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/19/01}\\
{\it Closed book \hfill Van Dommelen \hfill 10:35-11:25am}
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Show all reasoning and intermediate results leading to your answer, or
credit will be lost. One book of mathematical tables, such as Schaum's
Mathematical Handbook, may be used, as well as a calculator.

\begin{enumerate}

\item When numerically solving partial differential equations on a
  computer, normally a ``mesh'' of points must be defined.  Algebraic
  mesh generation requires that certain functions are chosen to define
  the mesh.  In one of my programs, I am using, in part, the following
  function:
  \begin{displaymath}
    f(x) = \ln\left(1+ C e^x\right)
  \end{displaymath}
  where $C$ is a constant.
  The reason I am using this function is that it has a horizontal
  asymptote for large negative $x$ and has a 45 degree asymptote for
  large positive $x$.  Do the following:
        \begin{enumerate}
        \item Verify the horizontal asymptote at large negative
        $x$ by finding $\lim_{x\to-\infty} f(x)$.  (The value
        of this limit is the height of the asymptote.)
        \item Find the $y$-intercept of the 45 degree asymptote.
        This intercept is given as the limit for $x\to\infty$ of
        the following function:
        \begin{displaymath}
                g(x)= f(x) - x = \ln\left(1+ C e^x\right) - x
        \end{displaymath}
        (Hint: you might want to find the limit of the
        function $e^g$ first.)
        (The fact that the intercept exists verifies that $f$
        does indeed have a 45 degree asymptote.)
        \end{enumerate}

\begin{rawhtml}
<A HREF="figures/1.gif">Solution.</A>
\end{rawhtml}

\item The rim angle $\alpha$ of a solar energy collector made up
out of plane mirrors satisfies the equation:
\begin{displaymath}
    \cos^2\left(\alpha\right)
        \left[1 + \sin\left(\alpha\right)
         - {\textstyle \frac12} \cos\left(\alpha\right)
        \right] = C
\end{displaymath}
where $C$ is a constant that depends on the desired concentration
factor, mirror density, height, etcetera.   Assuming that $\alpha$
is small, (i.e. that $C$ is close to a half), write an approximate cubic
equation for the rim angle $\alpha$.  (You do not have to solve
the cubic equation unless you want to.)

\begin{rawhtml}
<A HREF="figures/2.gif">Solution.</A>
\end{rawhtml}

\item Find the moment of inertia $\int x^2 \; {\rm d} A$ for
the flattened axis $x^2+y^2 \le R^2$ and $x\le \frac12 R$.

\begin{rawhtml}
<A HREF="figures/3.gif">Solution.</A>
\end{rawhtml}

\end{enumerate}

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