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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/26/07}\\
{\it Closed book \hfill Van Dommelen \hfill 11:50-12:40 pm}
\end{center}

Solutions should be fully {\em derived} showing all intermediate
results, using class procedures.  Show all reasoning.  Bare answers
are absolutely not acceptable, because I will assume they come from
your calculator (or the math handbook, sometimes,) instead of from
you.  You must state what result answers what part of the question if
there is any ambiguity.  Answer exactly what is asked; you do not get
any credit for making up your own questions and answering those.
Use the stated procedures.

One book of mathematical tables, such as Schaum's Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.

\begin{enumerate}
  
\item {\bf Background:} The volume of a carbon nano tube with
  spherical end caps can be approximated as
  \begin{displaymath}
    V=\pi r^2 \ell + \frac43 \pi r^3
  \end{displaymath}
  where $r$ is the radius of the tube and $\ell$ is the length.
  
  {\bf Question:} Find the expression for the time derivative of the
  volume when $r$ and $\ell$ change according to
  \begin{displaymath}
    \frac{{\rm d}r}{{\rm d}t} = r + \sin r
    \qquad
    \frac{{\rm d}\ell}{{\rm d}t} = \tan(r)
  \end{displaymath}

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Often, integrals are encountered for which an
  anti-derivative cannot be found.  Approximation is then needed.

  {\bf Question:} Approximate
  \begin{displaymath}
    \int_0^1 x\sin(x^3)\;{\rm d}x
  \end{displaymath}
  to an error no larger than 0.0005, doing the minimum amount of work
  possible.  Explain in detail how you can be sure that the error in your
  result is no larger than 0.0005.  Note: that is $\sin(x^3)$, not
  $\sin^3x$.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} If your laser beam is to move through a
  long pipe with flat sides, it must be parallel to the sides of
  the pipe.

  {\bf Question:} Find the equation of the line through point
  P, (1,2,3), and parallel to the planes
  \begin{displaymath}
    7 x + 4 y + 3 z = 5 \qquad 8 x + y - z = 21
  \end{displaymath}

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

\end{enumerate}

\end{document}
