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Problems PreCal 1508 PLTL Workshop, October 7, 2011\\
PL: Alex Knaust, Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}
\begin{enumerate}
\item Solve for $P$, then solve for $t$ in
\begin{enumerate}
\item $A = Pe^{rt}$
\item $A = P\left(1 + \frac{r}{n}\right)^{nt}$
\end{enumerate}

\item Use one of the above equations for compound interest to determine the principal $P$ that must be invested at a rate of $r = 5\%$, compounded monthly, so that \$500,000 will be available after $t$ years.

\item Complete the table for the time $t$ (in years) necessary for $P$ dollars to triple if interest is compounded continously at rate $r$. Describe the relationship between $r$ and $t$.
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\begin{tabular}{|c|c|c|c|c|c|}
\hline $r$ & 2\% & 4\% & 6\% & 8\% & 10\% \\ 
\hline $t$ &  &  &  &  &  \\ 
\hline 
\end{tabular}
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\item The amount of yeast in a brew of beer is increasing according to the law of exponential growth. ($y(t) = ae^{bt}$) After 3 hours, there are 100 yeast, and after 5 hours, there are 400 yeast. How many yeast will there be after 6 hours?

\item Which values of $x$ satisfy the equation?    $2\cdot 3^{-x +1} = 4e^{2x - 1}$

\item The age of artifacts can be estimated using Carbon-14 dating with following equation, where $R$ is the ratio of Carbon-14 to Carbon-12 in the artifact and $t$ is time
\[R = \frac{1}{10^{12}}e^{-t/8823}\]
\begin{enumerate}
\item What is the ratio of Carbon-14 in a fossil after 300 years?
\item The ratio of Carbon-14 to Carbon-12 in a piece of paper is $R = 1/13^{11}$. Estimate the age of the paper.
\end{enumerate}

\item Sketch the graph of the following equations shifted 3 units down and 2 to the right
\begin{enumerate}
\item $f(x) = e^{x + 1} - 2$
\item $g(x) = \ln{(x - 5)} + 2$
\end{enumerate}

\item Find Vertical and horizontal asymptotes, holes and intercepts of the rational function
\[ q(x) = \frac{2x^2 + 6x + 4}{x^2 + 5x + 4} \] 

\item Show that the following identities are true 
\begin{enumerate}
\item $\log_a{b} = \frac{1}{\log_b{a}}$
\item $(\log_a{b})\cdot(\log_c{d}) = (\log_c{b})\cdot(\log_a{d})$
\end{enumerate}
\emph{Hint : Use change of base}

\end{enumerate}
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