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\begin{center}
{\Large Final

Math 1090

Summer 2000
}
\end{center}
\vspace{.1in}

Each problem is worth 10 points. 
Show all your work and make your work legible.

\vspace{.1in}

Name \hrulefill

\vspace{.2in}

\begin{enumerate}

\item  Use the matrices to perform the operations.
If any operation is not possible, explain why.
\[
A = 
\left[
\begin{array}{rrr}
-1 & 0 & 6 \\
2 & 2 & -4 \\
\end{array}
\right]
\quad
B = 
\left[
\begin{array}{rr}
1 & 7 \\
0 & -1
\end{array}
\right]
\quad
C = 
\left[
\begin{array}{rr}
9 & -2 \\
1 & 0
\end{array}
\right]
\quad
D = 
\left[
\begin{array}{rrr}
2 & 1 & 0 \\
0 & 0 & -4 \\
1 & 1 & 1 
\end{array}
\right]
\]

\begin{enumerate}
\item  $AB$
\item  $BA$
\item  $3C-2B$
\item  $D^{-1}$
\end{enumerate}


\item  
\begin{enumerate}
\item  For $f(x)=3x^2-\frac{2}{5}x$, find and simplify the following:'
\[
\frac{f(x+2h)-f(x+h)}{h}
\]
\item  Find a decimal approximation for
\[
\log_{82}{9876}
\]
\end{enumerate}

\item  A firm has the following cost and revenue functions
\[
C(x)=3600+25x+\frac{1}{2}x^2 \quad \quad  R(x)=\left(175-\frac{1}{2}x \right)x
\]
\begin{enumerate}
\item  Is there a maximum or minimum profit (or neither)?
\item  Find this maximum or minimum profit (if there is one) and what level of production gives it.
\end{enumerate}


\item  You invest \$12500 at 8\% annual interest for 15 years.
\\ Find the future value of this investment, if
\begin{enumerate}
\item  the interest is compounded bi-annually,
\item  the interest is compounded monthly,
\item  the interest is compounded continuously.
\end{enumerate}


\item  Graph the inequalities and find all the corners.
Find the maximum of the function $f=2x+5y$ subject to the constraints:
\begin{eqnarray*}
-x+y & \leq & 4 \\
x+3y & \leq & 20 \\
5x+2y & \leq & 35 \\
x & \geq & 0 \\
y & \geq & 0 
\end{eqnarray*}


\item  Determine if the system of equations has a solution.  If it does, find it:
\begin{eqnarray*}
2x+2z & = & 1 \\
x-3y+4z & = & 2 \\
x+2y+z & = & 1
\end{eqnarray*}


\item  The percentage of high school seniors, $p$, who have tried marijuana 
can be modeled by
\[
p = 49.813783 + 2.7783t - 0.228596t^2
\]
where $t$ is the number of years past 1975.
In what year(s) did 27\% of high school seniors try marijuana?


\item  Suppose that the size $P$ of a deer herd $t$ years after being introduced into a region 
is given by
\[
P = 2200 - 2100(3^{-0.7t})
\]
How many years does it take the herd to reach a population of 1500?


\item  You take out a loan of \$8000 that you are to pay back over 5 years.
You make payments at the end of every 3 months.
The interest rate is 7\% annual interest compounded quarterly.
How much are the payments?


\item  Suppose your house payments are \$800 a month and your interest rate on your loan
is 7.5\% annual interest compounded monthly.
What is your balance of your loan if you have 10 year left to pay on your house?


\item  Suppose you start saving for your retirement by putting \$300 into an account at the
end of every two months.
This account pays 9\% annual interest compounded 6 times a year.
How much is this retirement fund worth in 20 years?


\item  
\begin{enumerate}
\item  Which of the following is better and why?
\begin{itemize}
\item  8.76\% annual interest compounded annually
\item  8.52\% annual interest compounded quarterly
\item  8.41\% annual interest compounded continuously
\end{itemize}


\item  Solve the equation
\[
\log 10^{2z} = 2(\log_5 5) -z^2 (\log_9 1) -4z
\]
\end{enumerate}
\end{enumerate}

\newpage

Formulas:
\[
\begin{array}{|l|c|}
\hline
& \\
\mbox{Arithmetic Sequence} &
	S_n=\frac{n}{2}\left(a_1+a_n\right) \\
& \\ \hline & \\
\mbox{Geometric Sequence} &
	S_n=a_1 \left(\frac{1-r^n}{1-r}\right) \\
& \\ \hline & \\
\mbox{Future Value} &
	S=P \left(1+\frac{j}{m}\right)^{mt} \\
& \\ \hline & \\
\mbox{Future Value} &
	S=P \left(1+i\right)^{n} \\
& \\ \hline & \\
\mbox{Effective annual rate} &
	r=\left(1+\frac{j}{m}\right)^{m} -1 \\
& \\ \hline & \\
\mbox{FV of Ordinary Annuity} &
	S=R \cdot \left[\frac{(1+i)^n-1}{i}\right] \\
& \\ \hline & \\
\mbox{FV of Annuity Due} &
	S=R \cdot \left[\frac{(1+i)^{n+1}-1}{i}\right] - R \\
& \\ \hline & \\
\mbox{PV of Ordinary Annuity} &
	A_n=R \cdot \left[\frac{1-(1+i)^{-n}}{i}\right] \\
& \\ \hline & \\
\mbox{PV of Deferred Annuity} &
	A_{(n,k)}=R \cdot \left[\frac{1-(1+i)^{-(n+k)}}{i} - \frac{1-(1+i)^{-k}}{i}\right] \\
& \\ \hline & \\
\mbox{Amortization} &
	R=A_n \cdot \left[\frac{i}{1-(1+i)^{-n}}\right] \\
& \\ \hline & \\
\mbox{Sinking Fund} &
	R=S \cdot \left[\frac{i}{(1+i)^{n}-1}\right] \\
& \\ \hline 
\end{array}
\]
\newpage






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