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\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Mock Midterm Exam}
\date{Monday, November 2, 2020}
\maketitle


\begin{enumerate}


\item Compute the following limits, if they exist.  Else, argue why
the limit does not exist.
\begin{enumerate}
\item $\displaystyle \lim_{s \to -1} \frac{\frac1s - 1}{s^3 - 1}$
\item $\displaystyle \lim_{x \to \infty} \frac{\e^{2x} + x^3 + \ln
x}{3 \, \e^{2x} - x^3 + \cos x}$
\item $\displaystyle \lim_{r \to 1} \frac{\lvert r-1 \rvert}{r^2-1}$
\end{enumerate}
\hfill(5+5+5)

\item The function $f (x)$ is defined on the interval $[0, 2]$ and is
between $4 - x$ and $x^2 + 2$ for all $x$ in this interval. Does it have to
be continuous at $x = 1$? Explain why or why not.
~~~~~~~~~~\hfill(5)

\item Show that the equation $x^7 - 3 \, x - 1 = 0$ has at least one
solution in the interval $[-1,1]$.
\mbox{~~~~~~~~~~}\hfill(5)

\item
\begin{enumerate}
\item Show that
\[
  \frac{\d}{\d x} \arctan x = \frac1{1+x^2} \,.
\]
\item
Consider the function
\[
  f(x) = 2 \, \arctan x - x \,.
\]
Find its domain, horizontal and vertical asymptotes, local minima,
local maxima, and inflection points of $f$.  Identify the regions
where the graph of $f$ is concave upward or concave downward.
Finally, sketch the graph of the function.
\end{enumerate}
\hfill(5+10)

\item An airplane is flying towards a radar station at a constant
height of 6\,km above the ground.  The distance $s$ between the airplane
and the radar station is decreasing at a rate of 400\,km/h when
$s=10$\,km.  What is the horizontal speed of the plane?
\hfill(10)
\newpage
\item Compute the following definite or indefinite integrals.
\begin{enumerate}

\item $\displaystyle \int x^{-3} \, \e^{1/x} \, \d x$

\item $\displaystyle \int \frac{x+1}{x^2 \, (x^2 + 1)} \, \d x$

\item $\displaystyle \int_0^{2\pi} (\cos^2 \phi - \sin^2 \phi) \, \d
\phi$

\end{enumerate}
\hfill(10+10+5)

\item Find the derivative of the function
\[
  F(x) = \int_{\sqrt x}^x \frac{\e^t}t \, \d t \,.
\]
\hfill(5)


\end{enumerate}

\end{document}
