\documentclass[12pt,fleqn]{article} \pagestyle{empty} \setlength{\topmargin}{-1in} \setlength{\oddsidemargin}{-.3in} \setlength{\textheight}{9in} \setlength{\textwidth}{7.2in} \begin{document} \begin{center} {\Large Final Math 1090 Fall 1998} \end{center} \vspace{.1in} Name \hrulefill \vspace{.2in} \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 $\log_2 10$ \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 end 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 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 \[ \begin{array}{ccccccc} x & + & y & + & z & = & 3 \\ 3x & - & 2y & + & 4z & = & 5 \\ x & + & 2y & + & z & = & 4 \end{array} \] \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 was 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 $2\log_3 x =2$ \item $t+2=\ln(e^{2t})$ \end{enumerate} \end{enumerate} \newpage \begin{center} {\Large Final Math 1090 - 2 Summer 1999} \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} \newpage \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} \newpage \begin{center} {\Large Math 1090 -- Final Exam Spring 2000} \end{center} \vspace{.1in} Name \hrulefill \vspace{.2in} Each problem is worth 10 points. Show all work. \vspace{.2in} \begin{enumerate} \item Use the matrices to perform the operations. If any operation is not possible, explain why. \[ A = \left[ \begin{array}{rr} -1 & 0 \\ 5 & 2 \\ 3 & -4 \end{array} \right] \quad B = \left[ \begin{array}{rr} 2 & 6 \\ 0 & -1 \end{array} \right] \quad C = \left[ \begin{array}{rr} 0 & -2 \\ 1 & -3 \end{array} \right] \quad D = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 0 & -2 \\ 0 & 1 & 1 \end{array} \right] \] \begin{enumerate} \item $AB$ \item $BA$ \item $4C-B$ \item $D^{-1}$ \end{enumerate} \item Total revenues of AT\&T can be modeled by \[ R = .253t^2 - 4.03t +76.84 \] where $R$ is in billions of dollars and $t$ is the number of years past 1980. In what year(s) was revenue \$70 billion? \item If total cost and revenue functions are \begin{eqnarray*} C(x) & = & 3600 + 100x +2x^2 \\ R(x) & = & 500x - 2x^2 \end{eqnarray*} find the maximum profit and what level of production gives this maximum. \item Find the maximum value of the function $z=5x+7y$ subject to the constraints. Make sure you graph the inequalities and find and label the corners. \begin{eqnarray*} -x+y & \leq & 4 \\ 3x-y & \leq & 15 \\ 2x+5y & \leq & 27 \\ x & \geq & 0 \\ y & \geq & 0 \end{eqnarray*} \item If $g(x)=2x^2-\frac{5}{3}x+1$, find and simplify the expression \[ \frac{g(x+h)-g(x)}{h} \] \item \begin{enumerate} \item Find a decimal approximation of \[ -(-0.3)^{\frac{2}{5}} \] \item Simplify: \[ \left( \frac{5x^{-2}}{3x^3y^{-1}} \right)^{-2} \] \item Given the following information: \[ \log_b 2 = 0.279 \quad \log_b 3 = 0.442 \quad \log_b 7 = 0.783 \] evaluate \[ \log_b \left( \frac{14}{3b} \right) \] \end{enumerate} \item You invest \$25,000 for 8 years at an annual interest rate of 9\%. \begin{enumerate} \item What is the future value if this is simple interest? \item What is the future value if interest is compounded every two months? \item What is the future value if interest is compounded continuously? \end{enumerate} \item Determine if the system of equations below has any solutions. If a solution exists, find it. \[ \begin{array}{rcrcrcr} 2x & + & 5y & + & 4z & = & 4 \\ x & + & 4y & + & 3z & = & 1 \\ x & - & 3y & - & 2z & = & 5 \end{array} \] \item Suppose that the size $y$ is a deer herd $t$ years after being introduced onto an island is given by \[ y = 2500-2490e^{-0.1t} \] How long until the herd reaches a population of $1500$? \item A brick patio has the approximate shape of a triangle. The patio has 21 rows of bricks, and these rows form an arithmetic sequence. The first row has 341 bricks and the $11$th row has 171 bricks. How many brick are in the patio. \item You buy a house and agree to make 50 quarterly payments of \$2500, with the first payment being at the end of the first quarter. If money is worth 10\% compounded quarterly, how much would the house have cost if you paid cash at the time of purchase? \end{enumerate} \newpage \begin{center} \textbf{Practice Final -- Spring 2000} \end{center} \hrule \vspace{.2in} \begin{enumerate} \item If you take a 30 year loan out of the bank for \$150,000, at 12\% interest compounded quarterly. What are your payments every quarter? \item Suppose you are making payments on a loan of \$185 a month for 5 years. You interest rate is 8\% compounded monthly. What was the original amount of the loan? \item Suppose you start a college fund for your newborn baby. You put \$600 into the bank twice a year, at the end of every 6 months. You interest rate is 6\% compounded bi-annually. What is this fund worth when you child turns 18? \\ What would the fund be worth if you deposited the money at the beginning of every 6 month period? \item Suppose you invest \$6000 for 5 years at 7\% annual interest. Find the future value of this investment if you: \begin{enumerate} \item Compound annually \item compound quarterly \item compound continuously \end{enumerate} \item Suppose you have an arithmetic sequence with 3rd term 12 and 6th term 21. Find a formula for the $n$-th term and find the sum of the first 30 terms. (Use the formula!) \item The number of people who know a rumor is given by the equation \[ N(t)=\frac{100,000}{5+200(2^{-.7t})} \] where $t$ is the number of days after the rumor starts. How many days until 15,000 people have heard? \item Find without a calculator \[ \log_{18} \left(\frac{1}{324}\right) \] \item Find a decimal approximation of $\log_8 16$ \item Sales revenues can be modeled by \[ R=-.037t^2+.782t+0.219 \] where $t$ is the number of years past 1980 and $R$ is in thousands of dollars. \begin{enumerate} \item Is there a maximum or minimum revenue? Find it. \item When is revenue equal to \$2,000? \end{enumerate} \item Find the minimum of $f=2x-7y$ subject to: (make sure you graph it and find the corners!) \begin{eqnarray*} x & \geq & 0 \\ y & \geq & 0 \\ 2x+y & \leq & 40 \\ x+5y & \leq & 100 \\ 8x+5y & \leq & 170 \end{eqnarray*} \item Given the matrices perform the operations (showing your work!): \[ A=\left[ \begin{array}{rrr} 0 & 2 & 1 \\ 3 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array} \right] \quad B=\left[ \begin{array}{rr} 2 & -2 \\ 0 & -3 \\ 1 & 0 \\ \end{array} \right] \] \begin{enumerate} \item $AB$ \item $BA$ \item $A-B$ \item $A^{-1}$ \item $B^{-1}$ \end{enumerate} \item Solve the system of equations: \begin{eqnarray*} x+2y-z & = & 3 \\ 2x+5y-2z & = & 7 \\ -x+y+5z & = & -12 \end{eqnarray*} \item Given supply and demand equations, find the break even point \begin{eqnarray*} \mbox{supply:} && p-4q=-9 \\ \mbox{demand:} && p+3q=26 \end{eqnarray*} \item Find the equation of the line perpendicular to $2x+7y=12$ through the point $(2,-2)$. \item Given $f(x)=4x^2-6x$ and $g(x)=2x-1$ find and simplify: \[ f\circ g(x)-g \circ f(x) \] \item Simplify \[ \left( \frac{\sqrt{1250x^{-6}}}{x^{-6}} \right)^{-1} \] \end{enumerate} \begin{center} \textbf{Practice Final - Answers (Spring 2000)} \end{center} A couple of these are wrong, but I can't remember which ones. (Sorry :-) ) \begin{enumerate} \hrule \item If you take a 30 year loan out of the bank for \$150,000, at 12\% interest compounded quarterly. What are your payments every quarter? This is a present value of an ordinary annuity. We look up the formula on the sheet and figure out that we need to find the payments $R$. We know that $A_n=150000$, $i=.12/4$, and $n=30\cdot 4=120$. We plug these all into the formula and solve for $R$: (or just use the amortization formula) \[ R=A_n \left( \frac{i}{1-(1+i)^{-n}} \right) = 4633.49 \] \hrule \item Suppose you are making payments on a loan of \$185 a month for 5 years. You interest rate is 8\% compounded monthly. What was the original amount of the loan? This is another present value of an ordinary annuity. Here we want to find the present value $A_n$. We know: $R=185$, $i=.08/12$, $n=12\cdot 5 =60$. we plug these in and \[ A_n = R \left( \frac{1-(1+i)^{-n}}{i} \right) = 9123.91 \] \hrule \item Suppose you start a college fund for your newborn baby. You put \$600 into the bank twice a year, at the end of every 6 months. You interest rate is 6\% compounded bi-annually. What is this fund worth when you child turns 18? This is future value of an ordinary annuity. We know that $R=600$, $n=18\cdot 2$, $i=.06/2$. We want to find future value: \[ S=R \left( \frac{(1+i)^n - 1}{i} \right) = 37965.57 \] What would the fund be worth if you deposited the money at the beginning of every 6 month period? This is now an annuity due, but otherwise the same. We just use a different formula: \[ S=R \left( \frac{(1+i)^{n+1} - 1}{i} \right) -R = 39104.53 \] \hrule \item Suppose you invest \$6000 for 5 years at 7\% annual interest. Find the future value of this investment if you: \begin{enumerate} \item Compound annually: Here we use the formula $S=P(1+i)^t$ with $P=6000$, $t=5$ and $i=.07$. This gives: $S=8415.31$ \item compound quarterly Here we use $S=P(1+i/n)^{nt}$ with $P=6000$, $t=5$, $i=.07$, and $n=4$. This gives: $S=8488.67$ \item compound continuously Here we use $S=Pe^{rt}$ with $P=6000$, $r=.07$ and $t=5$. This gives $S=8514.41$ \end{enumerate} \hrule \item Suppose you have an arithmetic sequence with 3rd term 12 and 6th term 21. Find a formula for the $n$-th term and find the sum of the first 30 terms. (Use the formula!) The formula for the general term of an arithmetic sequence is $a_n=a_1+d(n-1)$. The information we have been given is $a_3=12$ and $a_6=21$. Plugging this into the general formula gives the equations: \[ 12=a_1+d(2) \quad 21=a_1+d(5) \] This is a system of equations with two unknowns ($a_1$ and $d$). We solve it and get: $a_1=6$ and $d=3$. Thus the formula for the general term of this sequence is $a_n=6+3(n-1)$ To find the sum of the first 30 terms, we need to find the 30th term. We use the formula: $a_{30}=6+3(29)=93$. We now plug into the formula for the sum and get: \[ S_{30}=\frac{30(6+93)}{2} = 1485 \] \hrule \item The number of people who know a rumor is given by the equation \[ N(t)=\frac{100,000}{5+200(2^{-.7t})} \] where $t$ is the number of days after the rumor starts. How many days until 15,000 people have heard? We plug in $15000$ for $N$ and now need to find $t$. We first need to get the exponential stuff alone. I multiply by $5+200(2^{-.7t}$ and divide by $15000$. This gives the equation: $5+200(2^{-.7t}) = 100000/150000 (=20/3)$. Now, I subtract $5$ from each side and then divide by 200: $2^{-.7t} = 1/12$. Now log both sides to get the exponent down: $-.7t \ln 2 = \ln (1/12)$. Now divide both sides by $-.7\ln 2$: \[ t = \frac{\ln (1/12)}{-.7\ln 2} = 5.12 \] or a bit more then 5 days. \hrule \item Find without a calculator \[ \log_{18} \left(\frac{1}{324}\right) \] We need to solve the equation: $18^x=1/324$. We factor 324 and find that it is equal to $(18)^2$. Thus we must have $\log_{18} 1/324 = -2$. \hrule \item Find a decimal approximation of $\log_8 16$ This is a change of base: \[ \log_8 16 = \frac{\ln 16}{\ln 8} \approx 1.3333 \] (we could have done this without a calculator!) \hrule \item Sales revenues can be modeled by \[ R=-.037t^2+.782t+0.219 \] where $t$ is the number of years past 1980 and $R$ is in thousands of dollars. \begin{enumerate} \item Is there a maximum or minimum revenue? Find it. Maximum because the coefficient of $t^2$ is negative which means the parabola points down. We first find the $t$-coordinate of the vertex: \[ t=\frac{-b}{2a} = \frac{-.782}{2 \cdot (-.037)} = 10.567 \] Now, we plug this in to find the $R$ value: $R(10.567)=4.531$. So, the maximum revenue is $4.531$ thousand dollars (\$4531) which happens in 1990.567. \item When is revenue equal to \$2,000? We need to solve $-.037t^2+.782t+0.219=2$, or $-.037t^2+.782t-1.781=0$. Here we use the quadratic formula. \[ t = \frac{-.782 \pm \sqrt{(.782)^2-4(-.037)(-1.781)}}{2(-.037)} \] This gives $t=2.6$ or $18.5$. In other words, revenue is \$2000 in 1982.6 and 1998.5. \end{enumerate} \hrule \item Find the minimum of $f=2x-7y$ subject to: (make sure you graph it and find the corners!) \begin{eqnarray*} x & \geq & 0 \\ y & \geq & 0 \\ 2x+y & \leq & 40 \\ x+5y & \leq & 100 \\ 8x+5y & \leq & 170 \end{eqnarray*} I don't want to graph this, but look at number 9 on page 268. The corners are: \[ (0,0), (0,20), (20,0), (15,10), (10,18) \] Plugging these into $f$ gives a minimum of -140 when $x=0$ and $y=20$. \hrule \item Given the matrices perform the operations (showing your work!): \[ A=\left[ \begin{array}{rrr} 0 & 2 & 1 \\ 3 & 0 & 1 \\ 1 & 1 & 1 \end{array} \right] \quad B=\left[ \begin{array}{rr} 2 & -2 \\ 0 & -3 \\ 1 & 0 \end{array} \right] \] \begin{enumerate} \item \[ AB=\left[ \begin{array}{rr} 1 & -6 \\ 7 & -6 \\ 3 & -5 \end{array} \right] \] \item $BA$ is not possible because the dimensions are wrong. The number of columns of $B$ would have to be equal to the number of rows of $A$ in order to do this. \item $A-B$ is not possible because $A$ and $B$ would have to be exactly the same orders. \item $A^{-1}$: \begin{eqnarray*} A=\left[ \begin{array}{rrr|rrr} 0 & 2 & 1 & 1 & 0 & 0 \\ 3 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 1 & 1 & 0 & 0 & 1 \\ 3 & 0 & 1 & 0 & 1 & 0 \\ 0 & 2 & 1 & 1 & 0 & 0 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & -3 & -2 & 0 & 1 & -3 \\ 0 & 2 & 1 & 1 & 0 & 0 \end{array} \right] \\ \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & -3 & -2 & 0 & 1 & -3 \\ 0 & -1 & -1 & 1 & 1 & -3 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & -1 & -1 & 3 \\ 0 & -3 & -2 & 0 & 1 & -3 \end{array} \right] \\ \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & -1 & -1 & 3 \\ 0 & 0 & 1 & -3 & -2 & 6 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 1 & 0 & 3 & 2 & 5 \\ 0 & 1 & 0 & 2 & 1 & -3 \\ 0 & 0 & 1 & -3 & -2 & 6 \end{array} \right] \\ \leftrightarrow \left[ \begin{array}{rrr|rrr} 1 & 0 & 0 & 1 & 1 & -2 \\ 0 & 1 & 0 & 2 & 1 & -3 \\ 0 & 0 & 1 & -3 & -2 & 6 \end{array} \right] \end{eqnarray*} So, we can read off $A^{-1}$: \[ A^{-1} = \left[ \begin{array}{rrr} 1 & 1 & -2 \\ 2 & 1 & -3 \\ -3 & -2 & 6 \end{array} \right] \] \item $B^{-1}$ is not possible because only square matrices (same number of rows as columns) have inverses. \end{enumerate} \hrule \item Solve the system of equations: \begin{eqnarray*} x+2y-z & = & 3 \\ 2x+5y-2z & = & 7 \\ -x+y+5z & = & -12 \end{eqnarray*} Here, I prefer to use Gauss-Jordan elimination: \begin{eqnarray*} \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 3 \\ 2 & 5 & -2 & 7 \\ -1 & 1 & 5 & -12 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 3 & 4 & -9 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 4 & -12 \end{array} \right] \\ \leftrightarrow \left[ \begin{array}{rrr|r} 1 & 2 & -1 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -3 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|r} 1 & 2 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -3 \end{array} \right] \leftrightarrow \left[ \begin{array}{rrr|r} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -3 \end{array} \right] \end{eqnarray*} So, we have $x=-2, y=1, z=-3$. \hrule \item Given supply and demand equations, find the break even point \begin{eqnarray*} \mbox{supply:} && p-4q=-9 \\ \mbox{demand:} && p+3q=26 \end{eqnarray*} This is just a system of equations to solve. Again, I use Gauss-Jordan elimination: \begin{eqnarray*} \left[ \begin{array}{rr|r} 1 & -4 & -9 \\ 1 & 3 & 26 \end{array} \right] \leftrightarrow \left[ \begin{array}{rr|r} 1 & 0 & 1 \\ 0 & 1 & 5 \end{array} \right] \end{eqnarray*} In other words, the market equilibrium is when $p=1$ and $q=5$. \hrule \item Find the equation of the line perpendicular to $2x+7y=12$ through the point $(2,-2)$. We first find the slope of the given line. I do this by solving for $y$: $y = -(2/7)x-12/7$. Thus this line has slope $-2/7$. The slope of a line perpendicular to this is $m=7/2$. Now putting the point and this slope into the point slope form: \[ y-(-2)=\left( \frac{7}{2} \right) (x-2) \] Rewriting this: $y=(7/2)x-9$. \hrule \item Given $f(x)=4x^2-6x$ and $g(x)=2x-1$ find and simplify: \[ f\circ g(x)-g \circ f(x) \] \[ f \circ g(x) = f(g(x)) = f(2x-1) = 4(2x-1)^2 - 6(2x-1) = 16x^2-28x+10 \] \[ g \circ f(x) = g(f(x)) = g(4x^2-6x) = 2(4x^2-6x)-1 = 8x^2-12x-1 \] putting these together gives: \[ f\circ g(x)-g \circ f(x) = (16x^2-28x+10) - (8x^2-12x-1) = 8x^2-16x+11 \] \hrule \item Simplify \[ \left( \frac{\sqrt{1250x^{-6}}}{x^{-6}} \right)^{-1} \] Remember that negative exponents are ``on the wrong floor'': \begin{eqnarray*} \left( \frac{\sqrt{1250x^{-6}}}{x^{-6}} \right)^{-1} & = & \frac{(1250)^{-1/2}(x^{-6})^{-1/2}}{(x^{-6})^{-1}} = \frac{(1250)^{-1/2}x^3}{x^6} = \frac{1}{25 x^3 \sqrt{2}} \end{eqnarray*} \end{enumerate} \end{document}