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\begin{document}

\title{Applied Calculus}
\author{Midterm Exam II}
\date{October 15, 2015}
\maketitle

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% % \noindent
% % Useful identities:
% % \begin{gather*}
% %   \cos \theta = \frac{\e^{\i\theta}+\e^{-\i\theta}}{2} \qquad
% %   \sin \theta = \frac{\e^{\i\theta}-\e^{-\i\theta}}{2i} \\
% %   \cos^2 \theta = \tfrac12 \, (1 + \cos 2\theta) \qquad
% %   \sin^2 \theta = \tfrac12 \, (1 - \cos 2\theta) \vphantom\int \\
% %   \cos \theta \, \sin \theta = \tfrac12 \, \sin 2 \theta \vphantom\int \\
% %   \tilde f(\xi)
% %   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{-\i\xi x} \, f(x) \,
% %   dx \qquad 
% %   f(x)
% %   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{\i\xi x} \,
% %   \tilde f(\xi) \, d\xi \\
% %   \delta(x)
% %   = \frac1{2\pi} \int_{-\infty}^\infty \e^{\i\xi x} \, d\xi  \\
% %   c_k = \frac1L \int_{-L/2}^{L/2} \e^{-2 \pi \i k x/L} \, f(x) \, \d x
% %   \qquad
% %   f(x) = \sum_{k=-\infty}^\infty c_k \, \e^{2 \pi \i k x/L}
% % \end{gather*}

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\begin{enumerate}


\item Suppose that for a certain data set, the semilog graph
(base-$10$, logarithm on the $y$-axis) is a line through the points
$(1,-3)$ and $(2,-1)$.  Give an equation for $y$ as a function of $x$.


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\item Compute the derivative of the following functions.
\begin{enumerate}
\item $\displaystyle f(x) = (1+x)^2 $
\item $\displaystyle f(x) = \frac{a x}{b x + c}$ where $a$, $b$, and
$c$ are constants.
\item $\displaystyle f(x) = x^2 \, \e^{-x}$
\end{enumerate}
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\item The $\arctan$ function is a special function.  You will find the
following formula for its derivative in any table of mathematical
functions:
\[
  (\arctan x)' = \frac1{1+x^2} \,.
\]
Use this information to find the derivative of 
\[
  f(x) = \arctan \frac1{x^4} \,.
\]
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%\newpage 
\item Find the derivative of \[f(x) = \frac1x\] by explicitly computing
the limit of the difference quotient
\[
  f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}h \,.
\]
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%\newpage 
\item Find the equation of the tangent line at $x=0$ for the graph of
the function
\[
  k(x) = \ln (1 + r x)
\]
where $r$ is a given constant.
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%\newpage 
\item Consider the function 
\[
  f(x) = \frac1{2x} + \ln 2x
\]
For which values of $x$ is the function defined?  Find the vertical
and horizontal asymptotes (if any), find and classify all critical
points, determine where the function is concave up or concave down,
find all points of inflection, and sketch the graph.  

(State the explicit $y$ values for the special points you found only
where they are easy to compute.)
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\item You drive the 300\,km distance from Berlin to Hamburg at an
average speed of 100\,km/h.  Then you go on to Bremen, which is
100\,km from Hamburg, where there is a traffic jam on the A1 so that
you average speed drops down to 50\,km/h on this part of your journey.
What is your average speed over the entire distance?
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\end{enumerate}
\end{document}