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\begin{center}
Problems PreCal 1508 PLTL Workshop, October 25, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

\end{center}
These problems represent a review of the content that will be covered by the second exam. The answers
are available as a pdf at \url{http://alex.knaust.info/pltlfall2011/}
\begin{enumerate}

\item For the following functions, find their domain, horizontal, vertical and slant asymptotes, holes, as well as
$x$ and $y$ intercepts. Use that information to sketch a graph of the function\\[0.1in]
\begin{inparaenum}[a)]
\item $f(x) = \frac{x^2 - 5x + 4}{x^2 - 16}\quad\quad\quad$
\item $v(t) = \frac{t^2 - t - 2}{t^3 - 2t^2 - 5t + 6}\quad\quad\quad$
\item $h(x) = \frac{4x}{x^2 + 4}$
\end{inparaenum}

\item Graph the following functions
\begin{center}
\begin{inparaenum}[a)]
\item $f(x) = \log_3 (x - 4) + 2\quad\quad\quad$
\item $f(x) = e^{x + 3} - 5$
\end{inparaenum}
\end{center}

\item Simplify the following equations into one logarithmic or exponential expression\\[0.1in]
\begin{inparaenum}[a)]
\item $\log_3 x + 2\log_3 x\quad\quad$
\item $e^9 \cdot e^x\quad\quad$
\item $\ln 19 - \ln x\quad\quad$
\item $\frac{2^{x-3}}{4^3}\quad\quad$
\item $\frac{\ln(4x)}{\ln(5)}$
\end{inparaenum}

\item Solve for $x$ in the following equations\\[0.1in]
\begin{inparaenum}[a)]
\item $2^{x-3} = 32\quad\quad$
\item $e^{2x} - 6e^x + 8 = 0\quad\quad$
\item $\ln x - \ln 3 = 2$
\\[0.1in]
\item $\ln \sqrt{x + 8} = 3\quad\quad$
\item $\log_6(x+2) - \log_6 x = \log_6(x+5)$
\end{inparaenum}

\item Solve the following systems of equations using \textbf{Gau\ss-Jordan} elimination\\[0.1in]
\begin{inparaenum}[a)]
\item $ \left\{\begin{array}{lr}
	2x - y  +3z = 24\\
	 \quad\quad 2y -z =14\\
	 7x -5y \quad\quad = 6\\
	\end{array}
	\right.\quad\quad\quad$
\item
$ \left\{\begin{array}{lr}
	8x - 4y = 7\\
	5x + 2y = 1\\
	\end{array}
	\right.
	$
\end{inparaenum}

\item Solve the following systems of equations\\[0.1in]
\begin{inparaenum}[a)]
\item $ \left\{\begin{array}{lr}
	3x  -2y + z = 15\\
	-x  + y + 2z = -10\\
	x - y - 4z = 14	
	\end{array}
	\right.\quad\quad\quad$
\item $
\begin{bmatrix}
-1 & 0 & -3 \\ 
3 & -3 & -2 \\ 
2 & 5 & 1
\end{bmatrix} \cdot 
\begin{bmatrix}
x_1\\
x_2\\
x_3\\
\end{bmatrix}=
\begin{bmatrix}
-1\\
6\\
2\\
\end{bmatrix}
$
\end{inparaenum}

\item Perform partial fraction decomposition on the following expressions\\[0.1in]
\begin{inparaenum}[a)]
\item $\frac{1}{x^2 + x}\quad\quad\quad$
\item $\frac{x^4}{(x-1)^3}\quad\quad\quad$
\item $\frac{x^2 - 4x + 7}{(x+1)(x^2 -2x + 3)}$
\end{inparaenum}


\item For the following matricies 
\[
A = \begin{bmatrix}
1 & 0 & 2 \\ 
2 & 1 & 6 \\ 
-4 & 2 & -3
\end{bmatrix} 
\quad\quad\quad
B = \begin{bmatrix}
2 & 1 \\ 
1 & 4 \\ 
\end{bmatrix}
\quad\quad\quad
C = \begin{bmatrix}
1 & 2 \\
-1 & 3 \\
9 & 0 \\
\end{bmatrix}
\quad\quad\quad
D = \begin{bmatrix}
3 & 9\\
-5 & 4\\
\end{bmatrix}
\]
Find
\begin{inparaenum}[a)]
\item $3B - 2D\quad\quad$
\item $A \cdot C\quad\quad$
\item $C \cdot B\quad\quad$
\item $B^{-1}\quad\quad$
\item $D^{-1}\quad\quad$
\item $B \cdot D\quad\quad$
\item $\det(B)$
\end{inparaenum}






\end{enumerate}
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