%% Math 131, Fall, 2003
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\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Review Sheet for Midterm 3} \\
Selected Answers \\
Professor Hwang \\
\end{center}

\be

\item A student stands 20~feet from a long, straight wall holding a
  flashlight. As the student turns the flashlight, the spot moves
  along the wall. 

  \bd

  \item{(a)} Find an equation that relates the angle the beam makes
  with the perpendicular and the position of the spot, then use your
  answer to find the speed on the spot in terms of the angular speed
  with which the flashlight is turned.
  \bigskip

  \begin{center}
    \input{rsoln3a.eepic}
  \end{center}

  \soln Let $\theta$~be the angle of the flashlight beam from the
  perpendicular and $x$~the distance in feet along the wall. Then
  $x=20\tan\theta$, or
  $\frac{dx}{dt}=20\sec^2\theta\,\frac{d\theta}{dt}$. 

  \item{(b)} If the student turns the flashlight at 1~radian per
  second, find the speed of the spot along the wall.

  \soln When the beam makes angle~$\theta$, the speed of the spot is
  $20\sec^2\theta$ feet per second.

  \item{(c)} In part~(b), find the angle at which the spot is moving
  300~mph (440~ft/sec). How far from the student is the spot at this
  angle?

  \soln $\theta=\arccos\frac{1}{\sqrt{22}}\simeq 77.7^\circ$, $x\simeq
  91.65$~feet.

  \ed

  \sep

  \item A beam has rectangular cross section of width~$w$ and
  height~$h$. The strength of the beam (the maximum weight it can
  carry without breaking) is proportional to~$wh^2$. In this question,
  assume that a beam 4~inches wide and 8~inches high can carry
  30,000~lb (15~tons).

  \bd

  \item{(a)} Use linear approximation to estimate the weight that can
  be supported by a beam 4~inches wide and 9~inches high.

  \soln The height has increased by $12.5$\%; maximum load increases
  by about~25\%, to 37,500~lb.

  \item{(b)} Estimate the width of a beam 8~inches high that can
  support 37,500~lb.

  \soln A 25\%~increase in load requires about a 25\%~increase in
  width, or 5~inches.

  \item{(c)} List at least two physical assumptions you needed to make
  in order to answer this question.

  \soln All beams in question have the same length, are made of the
  same material, are tested at the same physical conditions
  (temperature, etc.)

  \ed

  \sep

  \item Use l'H\^{o}pital's rule to evaluate the following limits:
    \[
    \lim_{n\to\infty}n\ln\Bigl(1+\frac{x}{n}\Bigr)=x,\qquad
    \lim_{n\to\infty}\Bigl(1+\frac{x}{n}\Bigr)^n=e^x,\qquad
    \lim_{h\to0}(1+hx)^{1/h}=e^x.
    \]

  \item Arrange the following expressions in increasing order of size
  as $x\to\infty$:
  \[
  \frac{e^x}{x+e^x}\ll\ln x\ll\ln(e^x+1)\ll x^2\ll 1000x^{10}
  \ll e^{x^2} \ll e^{e^x}
  \]

  \sep

  \item Let $f(x)=x^{1/3}(1-x)$. Find and classify the critical points
  of~$f$. Does the graph of~$f$ have inflection points?

  \soln Critical points are $x=0$ (neither max nor min) and $x=1/4$
  (local min). There are two inflection points: $x=0$ and $x=1/2$.

  \sep

  \item Let $a>0$, $u$~a differentiable function, and
  $U(x)=e^{ax}u(x)$.

  \bd
  \item{(a)} Find a formula for the derivative of~$U$ in terms of $u$,
  $u'$, and~$a$.

    \soln $U'(x)=e^{ax}\bigl(u'(x)+au(x)\bigr)$

  \item{(b)} Use your answer to part~(a) to find the critical points
  of $f(x)=e^x\sin x$ and $g(x)=x^ne^{-x}$. Find the ``best possible
  bounds'' (see p.~184) for~$g(x)$, assuming $x\geq0$.
  \ed

  \soln $f'(x)=e^x(\cos x+\sin x)$, 
  $g'(x)=e^{-x}(nx^{n-1}-x^n)=e^{-x}x^{n-1}(n-x)$. Best bounds are
  $0\leq g(x)\leq n^ne^{-n}$.

  \sep

  \item A landscape architect is commissioned to enclose a 3000~square
  foot rectangular garden. The garden is to be bordered by shrubs on
  three sides and a fence along the fourth side; shrubs cost \$25/ft,
  while fence costs \$10/ft. Find the dimensions that minimize the
  total cost, and give the minimum cost.
  
  \soln The garden should be $10\sqrt{21}\times100\sqrt{3/7}$ feet, or
  about $45.6\times65.5$~feet.

\ee

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