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\begin{center}
{\Large Final

Math 1090 - 2}
\end{center}
\vspace{.1in}

Name: \hrulefill
\vspace{.2in}

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

\begin{enumerate}

\item 
\begin{enumerate}
\item If $f(x)=x^2-x$, find, and simplify, $\frac{f(x+h)-f(x)}{h}$.
\item  Get a decimal approximation of $\log_6 45$
\item Write the equation of the line through $(6, -4)$ that is parallel to
$4x - 5y = 6$
\end{enumerate}

\item  You invest \$25000 at 10\% per year annual interest, for 20 years.
\begin{enumerate}
\item  What is the future value if this is simple interest?
\item  What is the future value if you compound quarterly?
\item  What is the future value if you compound continuously?
\end{enumerate}

\item  Suppose that you just had a child.
The day that this child is born, you decide to set up a college fund.
At the beginning of each month you put \$200 in a savings account that 
gets 6\% annual interest, compounded every month.
How much is the college fund worth when the kid is 18 and ready to go to college?

\item  For your retirement you decide to put some money into an account that
earns 8\% annual interest compounded monthly.
You are going to withdraw \$500 a month for 12 years.
How much money do you need to deposit?

\item Given matrices:
\[ 
A=\left[ \begin{array}{lcr}
                        2 & 3 & 0  \\
                        1 & 2 & 5  
                     \end{array} \right]
\hspace{1in}
B=\left[ \begin{array}{lr}
                        0 & 2  \\
                        1 & 4 
                    \end{array} \right]
\]
\[ C=\left[ \begin{array}{lr}
                        -1 & 2 \\
                         2 & 3 
                    \end{array} \right]
\hspace{1in} 
D=\left[ \begin{array}{ccc}
1 & 2 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array} \right]
\]

Perform the indicated operations if possible. If an operation is not 
possible, explain why.
\begin{enumerate}
\item $BA$
\item $AB$
\item $B+C$
\item $4B$
\item $D^{-1}$
\end{enumerate}

\item Solve the system of equations
\[ 
\begin{array}{ccccccc}
3x & - & 2y & + & 4z & = & 5  \\
x & + & y & + & z  & = & 3  \\
x & + & 2y & + & z & = & 4
\end{array}  
\]

\item  The demand function for a product is given by
$p=100-4q-q^2$
and the supply is given by
$p=q^2+8q+20$.
Find the equilibrium quantity and price.

\item A hat maker has a little business that has fixed costs \$550 and 
additional costs of \$7 per unit.  
Each hat is sold for \$15 each.
Find
\begin{enumerate}
\item The profit function;
\item the marginal profit;
\item the break-even point.
\end{enumerate}


\item A small firm makes two types of watches.  The fancy watch takes 2 
hours and \$60 to make, and the calculator watch takes 3 hours and 
\$30 to make.  Suppose the firm has 120 hours and \$2,400 for the 
production of the watches.  
You want to find the maximum number of watches that can
be made each day.
\begin{enumerate}
\item Write down the inequalities that describe the above constraints.
\item Write down the correct equation maximize.

Make sure you tell what your variables represent.
DO NOT SOLVE the problem, just set it up.
\end{enumerate}


\item Given the following objective function $C=5x+4y$ subject to the 
constraints:
\begin{eqnarray*}
3x+2y & \leq  & 6 \\ 
2y-x  & \geq & 2 \\
x & \geq & 0
\end{eqnarray*}
\begin{enumerate}
\item Graph and shade the feasible region described by the constraints.
\item Solve for and label the corner points of part a).
\item Find the minimum value of the objective function $C$.
\end{enumerate}

\item Let $x$ be the amount (in hundreds of dollars) a company spends on 
advertising.  Assuming that the revenue $R$, and cost $C,$ are given by 
\[
R=100+41x-.5x^2
\hspace{1in}
C=225+5x+.25x^2
\]
what amount of 
advertising money will give the maximum profit?

\item A certain lake was stocked with 500 fish and the fish population 
$p(t)$ grew according to the logistics function 
\[p(t)=\frac{10,000}{1+19e^{-\frac{t}{5}}},\] where $t$ is the time in 
months since the lake was originally stocked.
\begin{enumerate}
\item Find the population after 30 months.
\item How long before the population is 7,000?
\end{enumerate}

\item The amount of aspirin in bloodstream of an initial amount of 
$500$ $mg$ after $t$ hours is given by 
\[ A(T)=500(0.944)^t.\]
How long does it take for the amount of aspirin dropped down to $200$
$mg$ in the bloodstream?


\item Solve the following equations.
\begin{enumerate}
\item $2\log_3 x = 15$
\item $t+2=\ln(e^{2t})$
\end{enumerate}





\end{enumerate}

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