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\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Review Sheet for Midterm 3}
Professor Hwang \\
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\paragraph{Coverage}

Midterm~3 will be given in class on Monday, November 21. The test
will cover the material from Sections 3.6~through 5.2 (except for
sections we skipped: 3.8, 4.4, 4.6, and~4.7).

In addition to the odd-numbered questions from the homework and the
questions on the first group worksheet, the problems below will
provide good practice for the actual test. The \emph{only} way to
prepare is to work lots of problems until you're fluent with the
techniques. Trying to memorize formulas is not a good study strategy.

There will be a review session from 6:30--8:00~PM on Thursday the~20th
in O'Neil~112. The midterms for the morning and noon sections will not
be the same; nonetheless, if you are in the morning section, it is not
to your advantage to discuss the test with students in the noon
section.

The important ideas and techniques from this portion of the course
include:

\bi

\item Related rates

\item Linearization and approximate linearity, l'H\^{o}pital's rule,
  asymptotic rates of growth (a.k.a.\ ``dominance'')

\item Geometric interpretations of the first and second derivatives,
  finding intervals on which a function is increasing/decreasing,
  concave up/down; critical points and inflection points, local and
  absolute extrema, the first and second derivative tests;
  optimization 

\ei

\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

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

  \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?

  \ed

  \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.

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

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

  \ed

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

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

  \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?

  \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~$a$.

  \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

  \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.

  \item p.~202, \#21 (If you can, find a calculus-free solution.)

  \item p.~203, \#23 

  \item p.~228, \#3 

  \item p.~228, \#7 

  \item p.~235, \#28 (If you can, give an argument that does not
  require a sketch.) 

\ee

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