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\LARGE{FINAL EXAM

Math 105 - 5}
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Show all your work and make your work legible.  Every problem is worth 10 points.  You may choose one problem to skip.

Skipped problem number: \rule{.5in}{.3pt}

\begin{enumerate}
\item Name \rule{5in}{.3pt}

\item  Determine if the following functions are odd, even or neither.
\begin{enumerate}
\item  $f(x)=17$
\vspace{1in}
\item  $g(x)=-3x^2+2x$
\vspace{1in}
\item  $h(x)=\sqrt[5]{2x}$
\end{enumerate}
\vspace{1in}

\item  $f(x)=x-3$, $g(x)=\sqrt{5-x}$.  Find:
\begin{enumerate}
\item  $\frac{f}{g}(x)$ 
\vspace{1in}
\item  Domain of $\frac{f}{g}(x)$
\end{enumerate}
%\vspace{1in}
\pagebreak

\item  Does $f(x)=\sqrt[3]{2x}$ have an inverse?  If it does find it.  If it does not, say why it does not have an inverse.
\vspace{2.5in}

\item  Solve
\[
|x^2+x|=2
\]
\vspace{2.5in}

\item  Solve and graph
\[\frac{x+12}{x+2} \geq 3\]
%\vspace{2.5in}
\pagebreak

\item  Find the least squares regression line $y=ax+b$ for the following data:
\[
\begin{array}{|c|c|}\hline
x & y \\ \hline
0 & 3 \\ \hline
1 & 6 \\ \hline
3 & 7 \\ \hline
4 & 10 \\ \hline
\end{array}
\]
\vspace{.2in}
You may find the equations helpful:
\begin{eqnarray*}
a = \frac{n \sum_{i=1}^{n} x_i y_i - \sum_{i=1}^{n} x_i \sum_{i=1}^{n}
y_i}{n \sum_{i=1}^{n} x_i^2 - (\sum_{i=1}^{n} x_i)^2}
 & &
b = \frac{1}{n} \left( \sum_{i=1}^{n} y_i - a \sum_{i=1}^{n} x_i \right)
\end{eqnarray*}
\vspace{2.5in}

\item  Find all zeros.  Write in factored form.
\[
f(x)=2x^3+x^2+x-1
\]
%\vspace{2.5in}
\pagebreak

\item  Graph, find zeros and all asymptotes.
\[
g(x)=\frac{x+1}{x-2}
\]
\vspace{3.5in}

\item  Solve
\[
\begin{array}{rrrrrrr}
          4x & - & y & + & 5z & = & 11 \\
          x & + & 2y & - & z & = & 5 \\
          x & - & 8y & + & 13z & = & 7 \\
          \end{array}
\]
%\vspace{3.5in}
\pagebreak

\item  Find the partial fraction decomposition 
\[
\frac{x-1}{(x+1)(x-2)}
\]
\vspace{3.5in}

\item  Graph the system of inequalities, find the max and min of $z=x-5y$
\[
\begin{array}{rrrrr}
       &  & x & \leq & 0 \\
       &  & y & \geq & 1 \\
     -2x & + & y & \leq & 5 \\
     \end{array}
\]
%\vspace{2.5in}
\pagebreak

\item  \label{matrix}  Find $AB$, $BA$, $A+B$  if possible.  If not possible, say why it is not possible.
\[
A=\left[ \begin{array}{rrr}
1 & -11 & 4 \\
8 & 6 & 4 \\
0 & 0  & 0 \\
\end{array} \right]  
B=\left[ \begin{array}{rr}
2 & 1 \\
-1 & 0 \\
1 & 1 \\
\end{array}\right]
\]
\vspace{3in}

\item Find $A^{-1}$ from problem~\ref{matrix} if possible.  If not possible say why not.
\vspace{2in}

\item  Find the area of a triangle with vertices at
$(-5,-6),(5,6),(-3,4)$
%\vspace{2.5in}
\pagebreak

\item  Find the sum:
\[
\sum_{i=1}^{100}\frac{i}{2}
\]
\vspace{2.2in}

\item  Find the sum:
\[
\sum_{n=0}^{\infty}5(\frac{1}{8})^{n}
\]
\vspace{2.2in}

\item  Find a formula for $a_n$ for a geometric sequence given that:  $a_1=16$ and $a_4=\frac{27}{4}$.
%\vspace{2.5in}
\pagebreak

\item  Prove using induction:
\[
2+7+12+\ldots +(5n-3)=\frac{n}{2}(5n-1)
\]
\vspace{4in}

\item  Find the term of $(z^2-2)^{15}$ with $z^{2}$ in it.
%\vspace{2.5in}


\end{enumerate}

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