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\title{Engineering and Science Mathematics 1A}
\author{Final Exam}
\date{December 13, 2010}
\maketitle

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Some trigonometric identities:
\begin{align*}
  {} &  \sin^2 \theta + \cos^2 \theta = 1
     && \sin 2x = 2 \, \sin x \, \cos x \\
     &  1 + \tan^2 \theta = \sec^2 \theta
     && \cos 2x = 2 \, \cos^2 x - 1 = 1 - 2 \, \sin^2 x \\
     &  1 + \cot^2 \theta = \csc^2 \theta
     && \tan 2x = \frac{2 \, \tan x}{1 - \tan^2 x} \\
     &  \sin^2 x = \frac{1 - \cos 2 x}{2}
     && \cos^2 x = \frac{1 + \cos 2 x}{2} 
\end{align*}

Useful integrals:
\begin{gather*}
  \int \frac{du}{\sqrt{1 - u^2}} = \arcsin u + C \\
  \int \frac{du}{1 + u^2} = \arctan u + C
                          = - \arccot u + C' \\
  \int \frac{du}{u \sqrt{u^2 - 1}} = \operatorname{arcsec} |u| + C \\
  \int \sec u \, du = \ln \bigl| \sec u + \tan u \bigr| + C  
\end{gather*}

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\begin{enumerate}

\item Compute the limits
  \begin{enumerate}
  \item $\displaystyle \lim_{n \to \infty} \frac{\sin n}{n}$
  \item $\displaystyle \lim_{x \to 0} \frac{1-\cos x}{x}$
  \item $\displaystyle \lim_{h \to 0} \frac{\sqrt{1+h}-1}h$
  \end{enumerate}
\hfill(5+5+5)
%\newpage

\item Prove that the equation
\[
  x^3 - 4x + 1 = 0
\]
has three (real) solutions.

\emph{Hint:} Don't try to compute the solutions!
\hfill (10)
%\newpage

\item Consider the function
\[
  f(x) = \frac{\ln x}x \,.
\]
What is the domain of $f$?  Find 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 into the coordinate system
provided.  \hfill(15)

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\item Compute the indefinite integrals
\begin{enumerate}
\item $\displaystyle \int \frac{x^2}{\sqrt{4-x^2}} \, dx$
\item $\displaystyle \int \frac{4x-2}{x^3-x} \, dx$
\end{enumerate}
\hfill (10+10) 
%\newpage

\item Compute the improper integrals

\begin{enumerate}
\item $\displaystyle \int_0^1 \frac{e^{\sqrt x}}{\sqrt{x}} \, dx$
\item $\displaystyle \int_0^{\tfrac\pi2}
       \frac{\sin x}{(\cos x)^{4/3}} \, dx$
\end{enumerate}
\hfill (10+10) 
%\newpage

\item
 Show, using integration, that the volume of a right circular
  cone of height $h$ and radius $r$ (see Figure) is given by \hfill(10)
\[
  V = \frac13 \, \pi \, h \, r^2 \,.
\]
\begin{flushright}
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\end{flushright}

%\newpage

\item Compute the Taylor series about the point $x=0$ of
  \begin{enumerate}
  \item $\displaystyle f(x) = \frac{1}{1-x}$
  \item $\displaystyle f(x) = \frac{1-x^2}{1-x}$
  \item $\displaystyle f(x) = \frac{1-x^3}{1-x}$
  \item Do you see a pattern?  Can you formulate a more general
  statement?
  \end{enumerate}
\hfill(5+5+5+5)
%\newpage

\item For which values of $p$ does the series
\[
  \sum_{n=2}^\infty \frac1{n \, (\ln n)^p}
\]
converge, respectively diverge?
\hfill (10)
%\newpage

\item Let $P(t)$ denote the number of fish in a lake at time $t$, and
  let $C$ denote the ``carrying capacity'' of the lake. Suppose
  further that fishermen catch a fraction $k$ of the fish per unit
of time, so that the population satisfies the equation
\[
  \frac{dP}{dt} = \Bigl( 1 - \frac{P}{C} \Bigr) \, P - k \, P 
  \qquad \text{with} \qquad
  P(0) = P_0 \,.
\]
\begin{enumerate}
\item For given values of $C$ and $k$, when is the population
  increasing, when is it decreasing?
\item For given values of $C$ and $k$, how many fish will be in the
  lake in the long run?  
(You do not need to solve the differential equation to answer this
question!) 
\item Which value of the  fishing rate $k$ maximizes the number of
  fish caught in the long run?
\item Solve the differential equation explicitly and check that your
  solution is
  consistent with your answer to part (b).
\end{enumerate}
\hfill(5+5+10+10)
\end{enumerate}
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