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\textbf{Final - Fall 99}

\begin{enumerate}

\item 
Use the matrices
\[
A = \left[
\begin{array}{rrr}
1  & 5 & 2  \\
-2 & 0 & -1 
\end{array}
\right]
\quad 
B=
\left[
\begin{array}{rr}
0 & 6  \\
2 & -1 
\end{array}
\right]
\quad 
C=
\left[
\begin{array}{rr}
1 & -2  \\
0 &  2
\end{array}
\right]
\quad 
D=
\left[
\begin{array}{rrr}
1 & 1 & 0  \\
0 & 0 & -2 \\
0 & 1 & 0  
\end{array}
\right]
\]
to perform the indicated operations if possible. If an operation is not 
possible, explain why.
\begin{enumerate}
\item $AB$
\item $BA$
\item $3B-2C$
\item $D^{-1}$ \quad (You are required to do this by hand, 
using row operations). 
\end{enumerate}

\item 
Determine if the system of equations below has any solutions. If a solution
exists, find it. Show all work.
\[
\begin{array}{rcrcrcr}
 x & +  & y   & +  & z  & = & 3 \\
3x & -  & 5y  & +  & 4z & = & 2 \\
 x & +  & 2y  & +  & z  & = & 4 
\end{array}
\]


\item  
If $f(x) = x^2+x$ find and simplify the expression
\[
\frac{f(x+h) - f(x)}{h}
\]

\item 
Find the equation of the line that passes through the point $(3,5)$ and is
parallel to the line $5x - 7y = 3$


\item
The total cost and total revenue functions for a product are 
\[
C(x) = 800 + 20 x \quad \mbox{ and } \quad R(x) = 100x - x^2
\]
respectively. Find the number of units $x$ that should be produced and sold in
order to  maximize the profit. What is the maximum profit?

\item 
For a certain product, the supply and demand functions are: 
\[
p = 2q-27 \quad \mbox{ and } \quad 2p+q=46
\]
respectively. Find the price and the quantity at the market equilibrium point.

\item 
We wish to maximize the function $f = 3x+2y$ subject to the constraints:
\begin{eqnarray*}
2x+y  & \leq & 8  \\
2x+3y & \leq & 12 \\
x     & \geq & 0  \\ 
y     & \geq & 0
\end{eqnarray*}
\begin{enumerate}
\item Sketch the feasible region, labeling all the corner points.
\item Where is the maximum value of $f$ attained? What is this maximum value?
\end{enumerate}

\item 
\begin{enumerate}
\item Find a decimal approximation of $\log_3 25$
\item Find $x$ if 
\[
\log_{16} x = \frac{3}{4}
\]
\end{enumerate}

\item The number of people in a town that have heard a rumor after $t$ days is
\[
N(t) = \frac{50,500}{1 + 500 e^{-0.4t}}
\]
How long does it take for 50,000 people to have heard the rumor?

\item 
The total revenue of a company can be modeled by the equation
\[
R(t) = 0.25t^2 - 4t + 85
\]
where the total revenue $R(t)$ is in billions of dollars and $t$ is the number
of years past 1980. For what values of $t$ will the total revenue equal \$75
billion?

\item 
A woman buying a house agrees to make 20 quarterly payments of \$1000 with the
first payment being at the end of the first quarter. If money is worth 8\%
compounded quarterly, how much would the house cost if she paid in cash at the
time of the purchase?

\item 
You invest \$50,000 for 10 years at an annual interest rate of 12\%.
\begin{enumerate}
\item What is the future value if this is simple interest?

\item What is the future value if interest is compounded quarterly?

\item What is the future value if interest is compounded
continuously?
\end{enumerate}

\end{enumerate}




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