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\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Review Sheet for Midterm 2}
Professor Hwang \\
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\paragraph{Coverage}

Midterm~2 will be given in class on Friday, October~31. The test
will cover the material from Sections 2.3~through 3.7, with emphasis
both on computation of derivatives and applications.

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 Wednesday
the~29th 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 The definition of the derivative, estimating a derivative from a
  table of data, interpretations of the derivative (as a rate of
  change, slope of tangent line, etc.), approximate linearity (the
  effect of zooming in on a differentiable function).

\item Geometric interpretations of the first and second derivatives,
  finding intervals on which a function is increasing/decreasing,
  concave up/down.

\item Rules for computing derivatives: The sum, product, quotient and
  chain rules, derivatives of inverse and implicit functions.

\item Specific derivatives: Power, exponential, trig and logarithm
functions.

\ei

\be

\item Let $f(x)= \sqrt{x}=x^{1/2}$ for $x>0$. 

  \bd

  \item{(a)} Compute the derivative of~$f$ at $x=4$ \emph{from the
  definition}. 

  \item{(b)} If you ``zoom in'' on the graph of~$f$ at the
    point~$(4,2)$, what will you see? Find the equation of the
    ``limiting line''.

  \item{(c)} If the height of the tangent line is used to estimate
  square roots, are the estimates over- or under-estimates? Explain.

  \item{(d)} Estimate $\sqrt{4+10^{-40}}$ to $60$~decimals. (You may
  write your answer in the form $2+\text{small number}$. Using the
  tangent line gives enough accuracy.)

  \ed

\item Let $g(t)=e^{-t^2/2}$. Find the interval(s) on which~$g$ is
  increasing, decreasing, concave up, and concave down. On which
  interval(s) is~$g$ both decreasing and concave down?  Use your
  answers to make an accurate sketch of the graph of~$g$ without using
  a graphing calculator or computer.

\item Find the equation of the line tangent to the curve $y^2=x^3-2x$
  at the point~$(2,-2)$.

\item Find the equation of the line tangent to~$y=e^x$ at $x=a$.

\item Let $f(x) = |x|(x-1)^{2/3}(x+1)^{1/3}$:
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  On the basis of the graph, does~$f$ appear to be continuous? At each
  of the labelled points, $f$~is not differentiable; sketch the
  limiting graph upon zooming in at the point.

\item Simplify $\ddx{x}(1-x)^5(1+x)^4$ and make a sign diagram for the
  derivative.

\item Assuming $f(x)=|x|x^n$, find a formula for~$f'(x)$.

\item Find the indicated derivatives; you may want to perform algebra
  before differentiating:

  \bd

  \item{(a)} $\ddx{u}\frac{1}{\ln u}$, \qquad $\ddx{x}x\ln x$, \qquad
    $\ddx{t}\ln(\ln t)$,\qquad $\ddx{x}\ln\frac{x-1}{x+1}$

  \item{(b)} $\ddx{x}\sqrt{x^2+1}$,\qquad
  $\ddx{z}\sqrt{\frac{z^2+1}{z^2-1}}$,\qquad
  $\ddx{v}\sqrt{e^v+e^{-v}}$

  \item{(c)} $\ddx{s}\frac{s^2+1}{\sqrt{s}}$,\qquad
  $\ddx{s}\frac{\sqrt{s}}{s^2+1}$, \qquad $\ddx{w}\frac{w^2-1}{w-1}$

  \item{(d)} $\ddx{\theta}\frac{\sin^2\theta}{\theta^2}$,\qquad
  $\ddx{y}\sin(y^2)$,\qquad $\ddx{r} r^2\cos r$,\qquad
  $\ddx{t}\arctan(\sin t)$

  \ed

\item An automobile costs $15,000$ and depreciates in value at
  18\%/year. Estimate the number of years that have passed when the
  rate of depreciation is \$1/day.

\item Suppose $f(x)=x^4 - 10x^3 + 3x^{5/2} -10$. Is it true that
  $f^{(5)}(x)=0$ for all~$x$? Explain. Find $f^{(4)}(x)$.

\ee

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