\magnification = \magstep1
\nopagenumbers
\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Sample Midterm Exam 3}
\centerline{November 19, 2001}
\bigskip
\noindent
{\it Note:  This is somewhat longer than the actual exam will be,
to give you an idea of the range of types of problems I could ask}
\bigskip
\noindent
I.  Compute the derivatives of the following:
\bigskip
\item{A)} $f(x) = \cos(3x^4 + x)$
\bigskip
\item{B)} $g(x) = {\ln(x)\over x^2 + 3}$
\bigskip
\item{C)} $h(x) = \arcsin(4x) + \arctan(2^x)$
\bigskip
\noindent
II. All parts of this question refer to the function 
$$f(x) = e^{-x}\cos(x).$$
\item{A)} Compute $f'(x)$ and $f''(x)$. 
\bigskip
\item{B)} Find the critical points $p$ of $f$ satisfying $-\pi \le p \le \pi$,
and classify each one as a local maximum, local minimum, or neither.
\bigskip
\item{C)} What are the global maximum and minimum values of 
$f(x)$ for $-1 \le x \le 1$ 
\bigskip
\noindent
III.  Because of security concerns, the Holy Cross Buildings and 
Grounds department wants to build a new enclosed parking lot for 
administrators.  To save costs they decide to make the lot by fencing 
in a rectangular area and using the existing fence at the non-enclosed 
end of Fitton Field (the end behind the scoreboard) as one of the four 
sides.  The area of the lot must be 500 square yards.  What should the 
dimensions of the lot be to minimize the length of new fencing needed?
\bigskip
\noindent
IV.  For these questions, say whether the statement is true or false.
If it is true, give a reason; if it is false, give an example of
a function showing that the statement is not true.
\item{A)}  If $f'(p) = 0$ and $f'$ is increasing 
at $p$, then $f$ has a local minimum at $x = p$.
\bigskip
\item{B)}  If $f'(p) = 0$, and $f''(p) = 0$, then
$f$ has neither a local maximum nor a local minimum at $x = p$.
\bigskip
\noindent
V.  Using the following information about $f'$, and $f''$ give
a qualitative sketch of the graph $y = f(x)$.
\bye
