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\centerline{\textbf{\large Exam \#3 $\cdot$ Thursday, November 6, 2008}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

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\textbf{Problem 1.} 
Consider the following table of values of a function $f$ and its first
two derivatives.
$$
\begin{tabular}{|r||r|r|r|r|r|}
\hline $x$      & $-3$   & $-1$  & $1$   & $3$   & $5$    \\
\hline
\hline $f(x)$   & $3.0$  & $4.0$ & $5.0$ & $8.0$ & $9.0$  \\
\hline $f'(x)$  & $1.0$  & $0.0$ & $0.5$ & $1.0$ & $0.3$  \\
\hline $f''(x)$ & $-2.0$ & $0.0$ & $1.4$ & $0.0$ & $-0.4$ \\
\hline
\end{tabular}
$$
Furthermore, $f'$ and $f''$ have no zeroes other than the ones shown.

\textbf{Part (a).}
Does $f$ have a critical point at $x=-1$?  Why or why not?

% True:  definition of critical point.
\vspace*{1.1in}

\textbf{Part (b).}
Does $f$ have a local maximum at $x=-1$?  Why or why not?

% False: f' positive on either side; fails the first derivative test.
% The second derivative test is inconclusive since f''(-1) is 0.
\vspace*{1.1in}

\textbf{Part (c).}
Does $f$ have an inflection point at $x=3$?  Why or why not?
\vspace*{1.1in}

% True:  f'' changes sign at x=3.

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\textbf{Problem 2.}
Consider the curve
$$
	\ln(xy) = 2x.
$$

\textbf{Part (a).}  Find $dy/dx$.
\vspace*{1.5in}

\textbf{Part (b).} Find an equation for the tangent line the curve at the point
$(x,y)=(1,e^2)$.

%\vspace*{1.75in}

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\newpage
\textbf{Problem 3.}

\textbf{Part (a).}
Find $B$ so that
$$
	G(x) = B 2^x + 2^{-x}
$$
has a critical point at $x=-1$.
\vspace*{2.5in}

\textbf{Part (b).}  Is this critical point a minimum, maximum, or neither?
Explain your reasoning.
\vspace*{2.0in}

%\vspace*{2.5in}

%%% ----------------------------------------------------------------
%\textbf{Problem 4.}
%Let $f(t) = te^{-at}$.
%Find $a$ such that $f(t)$ has an inflection point at $t=7$.

%\vspace*{0.2in}

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\textbf{Problem 4.}
For each of the following, does the limit exist?
If so, what is it, and why?  If not, why not?
\vspace*{0.1in}

\textbf{Part (a).}
$
	\lim_{t\to 0} \frac{1-\cosh t}{t^2}
	\qquad
	\qquad
	\qquad
	\qquad
$
\vspace*{2.1in}

\textbf{Part (b).}
$
	\lim_{z\to 0} \frac{3^z}{z^3}
$
%\vspace*{1.2in}

%\textbf{Part (c).}
%$
%	\lim_{z\to 0} \frac{z^3}{3^z}
%$
%
%\textbf{Part (d).}
%$
%	\lim_{x\to -\infty} \frac{1 + 4x + 2x^2}{-7 - 3x^2}
%$

%\vspace*{0.2in}

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\newpage
\textbf{Problem 5.}
The east and west sides of a rectangular enclosure cost \$50 per meter; the
north and south sides cost \$80 per meter.  Find the dimensions of the
enclosure with least cost enclosing an area of 1000 square meters.

\vspace*{4.5in}

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\textbf{Problem 6.}
The trajectory of an orbiting object is described by
$$
	r(1 + 0.2\cos(\theta)) = 10.
$$
(The units of $r$ are thousands of kilometers, or megameters if you like, and
the units of $\theta$ are radians.) Find $d\theta/dt$ when $\theta=\pi/3$,
given that $dr/dt = -20$ megameters per hour when $\theta=\pi/3$.  Compute your
answer to three decimal places.

%\vspace*{0.2in}

\end{document}
