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\begin{document}


\begin{center}
\textbf{Math 191\quad Spring 1998}

\textbf{Assignment 5: More Derivative Stuff}
\end{center}

\textbf{The due date for this assignment is Friday February 11.}

Reading assignment: Sections $3.4$, $3.5$, $3.6$, $3.S$, $4.3$.

\begin{enumerate}
\item Los socios comerciales mas importantes de Venezuela son Estados Unidos
y Colombia. Como ustedes saben San Antonio del Tachira y C\'{u}cuta son
ciudades vecinas en la frontera y adem\'{a}s son dos mercados donde se
desarrolla un considerable mercado cambiario donde prevalecen el d\'{o}lar,
el Bol\'{\i}var y el peso. Actualmente, Venezuela tiene un sistema de
control de cambios donde la tasa oficial de cambio es de Bs/US\$2,15. Los
Vene

\item Venezuela's most important trading partners are the U.S. and Colombia
(its neighbor). San Antonio (Venezuela) and Cucuta (Colombia) are neighbor
cities in the Venezuelan-Colombian frontier and where a remarkable amount of
trade between Bolivars, Pesos, and dollars takes place. Currently, Venezuela
has an official exchange rate (under exchange controls) of 2.15 Bolivars per
U.S. Venezuelans can use up to 5,000 dollars per year (at the official rate)
with their credit cards when outside the country. In Colombia, the free
market exchange rate is 1,945 Pesos per dollar and 410 pesos per Bolivar.

\item a.\qquad Assume you live in San Antonio (Venezuela) and you have 2,150
Bolivars, can you make a profit in the foreign exchange market? Why or why
not? Show your work.

\item b.\qquad Assume now that you live in Cucuta (Colombia) and you have
1,945,000 Pesos. Can you make a profit in the foreign exchange market? Why
or why not? Show your work.

\item c.\qquad What currency, if any, is undervalued (overvalued)? If you
think one of the currencies is undervalued (overvalued), what do you think
the exchange rates should be?

\begin{description}
\item[(a)] $f(z)=z^{3}+2z+1$

\item[(b)] $g(x)=1/(3x^{2})$

\item[(c)] $\gamma (t)=\sqrt[3]{t}$
\end{description}

\item Use problem 1 from assignment 3 and the definition of derivative to
show that $(f+g)^{\prime }=f^{\prime }+g^{\prime }$.

\item Do the following exercises from the text:

\begin{itemize}
\item page 108 numbers 3 to 18 multiples of 3

\item pages 115 and 116 numbers 9 to 36 multiples of 3

\item pages 101 and 102 numbers 3 to 30 multiples of 3

\item pages $120$ and $121$ numbers $9$ to $36$ multiples of $3$

\item page $144$ numbers $3,6,8,9.$
\end{itemize}

\item Review the concept of inverse functions from Chapter $1$.

\begin{description}
\item[(a)] Suppose $f$ and $g$ are inverse functions and $g^{\prime }(x)$ is
known. Use chain rule and the fact that $f$ and $g$ are inverse functions to
find $f^{\prime }(x)$ in terms of $g^{\prime }$ and $f.$

\item[(b)] Suppose $E(x)$ and $L(x)$ are inverse functions and $L^{\prime
}(x)=\frac{1}{x}$. Use the above result to find $E^{\prime }(x)$.
\end{description}

\item Simultaneous tangent lines. Sketch the graphs of $f(x)=-(x^{2}+1)$ and 
$g(x)=(x-1)^{2}$. Find the equations of the lines that are tangent to both
curves simultaneously.
\end{enumerate}

\end{document}
