\documentclass[12pt,fleqn]{article} \setlength{\topmargin}{-1in} \setlength{\oddsidemargin}{-.3in} \setlength{\textheight}{9in} \setlength{\textwidth}{7.2in} \begin{document} \begin{center} {\Large Final Review Math 1090 - 4} \end{center} \hrule \vspace{.2in} The final is: Thursday, December 16, 4-6pm, in \textbf{JTB 130} I plan to update this as I get more information. Check back from time to time. \begin{enumerate} \item I will have office hours on Wednesday Dec 15 at: \rule{2in}{.2pt} \item Sample problems at: \textrm{www.math.utah.edu/\~{}thornton/teaching/m1090/} \item You need to know how to write your answer in a sentence! Make sure you show \underline{all} your work when you take the final. \item Chapter 1. Know how to find equations of lines and what parallel and perpendicular lines are (1.7:1-40). Understand functions and how to compose two functions (1.3:7-16 and handout). Understand cost, revenue, profit, and supply and demand. This includes break-even analysis and market equilibrium (1.6: 1-24,1.9: 1-38). Know how to solve systems of linear equations, you can and should use methods from chapter 2. (1.8:all especially 1-22) \item Chapter 2: Matrices. Know how to add, and multiply matrices (2.1:15-26, 2.2:1-42). Know elimination (2.3:1-26). Know how to find inverse matrices and know how to use them (2.4:15-22,29-36). \item Chapter 3: Inequalities. Know how to solve and graph these in 2 variables (3.2:17-32), and how to do linear programming(3.3:13-53). Because a linear programming problem will demonstrate the ability(or lack thereof) to solve and graph a a system of inequalities, you can expect only a linear programming problem on the final (although its not set in stone yet). There will be no problems involving the simplex method on the final. \item Chapter 4: quadratic functions. Know the quadratic formula and how to use it (4.1:49-63). Know how to find the maximum or minimum of a quadratic function (4.2:35-46,50-53). Understand the applications in this chapter. This should just be a matter of understanding break even analysis and marker equilibrium and what to do when quadratic functions are involved (4.3:all). \item Chapter 5: logarithms and exponentials. The new thing here was logarithms. Understand how to convert a log equation into a exponential equation and vice-versa (5.2:1-12). Know the properties of logarithms (5.2:19-36, if it was up to me, I would test the understanding of these by using a story problems somehow, but you do need to know this). The most important thing in this chapter is solving equations (5.3:all). Another possibility that could be tricky without practice is some problems like (5.review:23-26). \item Chapter 6: finances. You will have the formulas for annuities on the test, but you will have know which formula to use and how to use it. For example, if its an annuity and you deposit at the end of the period you use an ordinary annuity. If you deposit in the beginning of a period, you use an annuity due. You need to know the difference between simple interest, compounded interest, and interest that is compounded continuously. You must know all the types of annuities in the book (future and present value). How do you tell the difference between a future value of an annuity and a present value of an annuity? (6.1:1-16, 6.2:1-40, 6.3:1-48). \end{enumerate} \newpage \hrule \vspace{.2in} Final fall 98: \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 \hrule \vspace{.2in} Old tests from fall 98: \begin{enumerate} \item The following function is used to determine a persons water bill: \[ C(x) = \left\{ \begin{array}{ll} 1.5x & 0 \leq x \leq 100 \\ 1.5x +1.2(x-100) & 100 < x \leq 1000 \\ 1000 + .7(x-1000) & x > 1000 \\ \end{array} \right. \] Where $C(x)$ is the cost in dollars for $x$ thousand gallons of water. \begin{enumerate} \item What is $C(200)$? \item Write a sentence explaining what $C(200)$ means. \end{enumerate} \item Find the equation of the line through the points: $(2,5), (5,-3)$. \item You decide to start a T-shirt business. Just starting out, you only offer one type of T-shirt and that will sell for \$15 each. You make the T-shirts at your home so there is no building to rent, but you do set up a new phone line, a 1-800 number and a 1-900 number. This phone line costs \$115 per month. You are able to produce each shirt for \$7 each. \begin{enumerate} \item What is the cost function $C(x)$ for this business? \item What is the revenue function $R(x)$ for this business? \item What is the Profit function $P(x)$? \item If you sell 10 shirts, do you make or lose money, and how much? \end{enumerate} \item In your T-shirt business of previous problem you figure that your business worth can be modeled by the equation: \[ W=2500+1200t \] where $W$ is the worth of the business in dollars, and $t$ is the number of years past 1998 (you start the business in January 1998). In what year do you project your business to be worth \$9100? \item If $f(x)=x^2-x$, find, and simplify, $\frac{f(x+h)-f(x)}{h}$. \item If $f(x)=\frac{1}{x-1}$ and $g(x)=x^3-x^2$, find, $f\circ g(x)$. (You don't need to simplify it). \item Graph the line $3x-2y=6$ \item Given the two functions: find the composition: (don't simplify it) \[ f(x)=3x^2-5x \] \[ g(x)=\sqrt{x-5} \] Find the composition $g \circ f(x)$ (don't simplify it) \item Write the function as a composition. (Find $f$ and $g$ so that $R(x)=f(g(x))$) \[ R(x)=\sqrt[3]{-2x^3-\sqrt{x}} \] \item Solve the system of equations: (use any method you wish) \[ \begin{array}{ccccccc} x & + & 3y & - & 8z & = & 20 \\ & & y & - & 3z & = & 11 \\ & & 2y & + & 7z & = & -4 \\ \end{array} \] \item Suppose that you are selling veggie burritos at the upcoming Phish concert. You can sell a burrito for \$5. You have fixed costs of \$50 (cooler, gas, ticket, parking), and each burrito costs you \$2 to produce. How many burritos must be sold in order to break even? \item Find, if possible. If not possible state why not. \[ A=\left[ \begin{array}{rrr} 3 & 0 & 1 \\ -2 & 1 & 0 \end{array} \right]\hspace{.5in} B= \left[ \begin{array}{rr} 2 & -1\\ 3 & -2 \\ -5 & 7 \\ \end{array} \right] \] \begin{enumerate} \item $2A-3B$ \item $AB$ \item $BA$ \end{enumerate} \item Consider the following system of equations. \[ \begin{array}{ccccccc} x & - & 3y & + & 3z & = & 7 \\ 3x & + & 2y & + & 4z & = & 5 \\ x & + & 2y &- & z & = & -2 \\ \end{array} \] \begin{enumerate} \item Write the augmented matrix for the system of equations. \item Using the ``legal operations'' on the matrix, how would you make the $(2,1)$ position a 0? \item Make the $(2,1)$ position a 0 as described above. \item If you were working with the equations, what would you have just done? (Explain what making the $(2,1)$ position means in terms of $x,y,z$.) \end{enumerate} \item Find the inverse of the matrix below \[ A=\left[ \begin{array}{rrr} -1 & 2 & 1 \\ -5 & 8 & 2 \\ 7 & -11 & -3 \end{array} \right] \] \item Are the two matrices below inverses? \[ \left[ \begin{array}{rrr} 2 & -1 & -2 \\ 3 & -1 & 1 \\ 1 & 1 & -1 \end{array} \right] \hspace{.5in} \left[ \begin{array}{rrr} 0 & 1/4 & 1/4 \\ -1/3 & 0 & 2/3 \\ -1/3 & 1/4 & -1/12 \end{array} \right] \] \item Solve the system of equations \underline{using} inverse matrices. \[ \begin{array}{ccccccc} 2x & - & y & - & 2z & = & 7 \\ 3x & - & y & + & z & = & 5 \\ x & + & y & - & z & = & -2 \\ \end{array} \] \item You are working at Gray Whale and sell a bunch of cd's. You sell some cd's at \$8 each and some at \$9 each. You end up selling 87 cd's and getting \$732. How many cd's do you sell at \$8 and how many at \$9? (You will \underline{not} get credit without setting up the equations.) \item For a certain product the revenue function is $R(x)=40x$ and the cost function is $C(x)=20x+1600$. When is the revenue greater then the profit? \item Graph the solution to the inequalities and find all the corners of the shaded region. \begin{eqnarray*} y - 2x & \leq & 0 \\ 2y - x & \leq & 4 \\ y & \geq & 2 \\ y & \leq & 6 \end{eqnarray*} \item Find the maximum and minimum of the function $A=3x-2y$ when $x$ and $y$ are in the shaded region of the previous problem. \item You spend some time researching VW camper vans and studying their acceleration. You test the following model: The time $t$, in \emph{minutes}, that it takes a Volkswagen camper van to accelerate to $x$ miles per hour can be modeled by the equation: \[ t = .01(-597.62+.1976x^2 +21.017x) \] How fast can the camper van go after 11 minutes? (Your answer may not reflect the acceleration in your car, but remember that this is a VW.) \item The percentage of (U.S.) workers that belong to unions from 1930 to 1989 can be modeled by the following equation where $t$ in in years past 1900, and $u$ is the percentage of workers belonging to unions in that year. \[ u = -39.3213 + 2.38432t - .022129 t^2 \] Did the percentage of workers belonging to unions reach a maximum or a minimum? When did this occur, and what was the maximum or minimum percentage? \item Graph the function. Find the coordinates of the vertex and label it. Find the zeros and label them too. \[ y=-2(x^2+x)-8x-13 \] \item You and a friend are making bumper stickers. You make 2 varieties: simple ones, and fancy ones. The simple sticker takes 15 minutes to make and the fancy ones take 30 minutes to make. You sell the simple stickers for \$2 each and the fancy ones for \$3 each. Between the two of you, you have 30 hours total per week to make stickers. The sticker market is such that you can only sell a total of 75 stickers each week. How many of each stickers should you make of each type to maximize your revenue? (Hint: I would convert to hours first. To get full credit you have to find the right inequalities, and graph the inequalities.) \item You are making bumper stickers again. You have fixed costs of $\$10$. If $x$ is the number of stickers you make, your cost per sticker is $2-\frac{1}{10}x$. Your selling price per sticker is $5-\frac{1}{5}x$ \begin{enumerate} \item What is the cost function? $C(x)=$ \item What is the revenue function? $R(x)=$ \item What is the profit function? $P(x)=$ \item What is the break even point(s). (Answer in a sentence!) \item (Extra credit-1 point) What is maximum or minimum profit? \end{enumerate} \item Your bumper sticker business lasts a while and you model the number of employees you have by the equation \[ N=-.23t^2+4.89t+52.12 \] where $t$ is the number of years past $1998$ and $N$ is the number of employees. Do you have a maximum or minimum number of employees. When does this happen, and how many employees? (A good sentence!!!) Extra credit-1 point: When does your business die how do you know it dies then? \item Solve the equation using factoring and methods learned in class. (Hint: $-\frac{5}{2}$ is a solution.) \[ 4x^4+20x^3+15x^2-29x-10 \] \item \begin{enumerate} \item Write the following equation in logarithm form \[ 5^{-3}=\frac{1}{125} \] \item Write the following equation in exponential form \[ \log_3 \frac{1}{9}=-2 \] \end{enumerate} \item Get a decimal approximation of $\log_2 10$, and check your answer using exponentials. (Show your work.) \item The length $x$, in centimeters, of brook trout is modeled by the equation \[ x=60-50e^{-.05t} \] where $t$ is the age of the fish in months. You catch a 55~cm trophy brook trout. How old is it? (Sentences!!) (I'll give you 1 point extra credit if you have the exact answer too.) \item Volkswagen camper owners have found that the more stickers on their buses, the more likely they are to be pulled over by the police and illegally searched. It is somehow determined that if you drive your VW bus from coast to coast, with $x$ stickers on your bus, the chance $P$, in percentage, that you will be pulled over and illegally searched at least once is \[ P=90 (.12)^{.91^x} \] \begin{enumerate} \item What is your chance of being illegally searched with no stickers on your bus? (Sentences!!) \item How many stickers can you have on your bus if you are willing to accept a $30\%$ risk of being illegally searched. (I'll give you 1 point extra credit if you also have the exact answer.) (Sentences!!) \end{enumerate} \end{enumerate} \newpage \hrule \vspace{.2in} Old tests from summer 99: \begin{enumerate} \item The following function is used to determine a persons water bill: \[ C(x) = \left\{ \begin{array}{ll} 1.5x & 0 \leq x \leq 100 \\ 1.5x +1.2(x-100) & 100 < x \leq 1000 \\ 1000 + .7(x-1000) & x > 1000 \\ \end{array} \right. \] Where $C(x)$ is the cost in dollars for $x$ thousand gallons of water. \begin{enumerate} \item What is $C(1000)$? \item Write a sentence explaining what $C(1000)$ means. \end{enumerate} \item Find the equation of the line through the points: $(2,5), (5,-1)$. \item You decide to start a T-shirt business. Just starting out, you only offer one type of T-shirt and that will sell for \$15 each. You make the T-shirts at your home so there is no building to rent, but you do set up a new phone line, a 1-800 number and a 1-900 number. This phone line costs \$115 per month. You are able to produce each shirt for \$7 each. \begin{enumerate} \item What is the cost function $C(x)$ for this business? \item What is the revenue function $R(x)$ for this business? \item What is the Profit function $P(x)$? \item If you sell 10 shirts, do you make or lose money, and how much? \item What is the break even point? \end{enumerate} \item Lets say you are car shopping. You are dead set on a new beetle. Let say that Volkswagen sells the beetle to Dave Strong (VW dealer) and 20\% of that price is VW's mark-up. Then, on top of that, 30\% of the price you pay is Dave Strong's mark-up. If the Beetle costs VW \$9250, how much do you have to pay for your Beetle? (Notice that there are two mark-ups going on here.) \item If the supply for a certain product is given by $2p-q=5$ and the demand is given by $3p+q=60$. What is the market equilibrium point? \item If $f(x)=\frac{x}{x^2-1}$ and $g(x)=x^3-x^2$, find, $f\circ g(x)$. (You don't need to simplify it). \item For the function $h$ below, find $f$ and $g$ so that $h(x)=f(g(x))$. \[ h(x)=\frac{52}{\sqrt{5x^{4}-6\sqrt{x}}} \] \item SET THIS UP, BUT DO NOT SOLVE IT. Make sure you tell me what every variable you use stands for. An airline company has three type of planes that carry three type of cargo. The payload of each plane is: \[ \begin{array}{c|ccc} \mbox{type of cargo} & \mbox{Passenger plane} & \mbox{Transport plane} & \mbox{Jumbo Plane} \\ \hline \mbox{Mail} & 110 & 90 & 120 \\ \mbox{Passengers} & 140 & 25 & 375 \\ \mbox{Air Freight} & 50 & 60 & 30 \end{array} \] Suppose that the company needs to move exactly 1200 units of mail, 2910 passengers, and 760 units of air freight. How may aircraft of each type is needed? \item Find, if possible. If not possible state why not. \[ A=\left[ \begin{array}{rrr} 3 & 0 & 1 \\ -2 & 1 & 0 \end{array} \right]\hspace{.5in} B= \left[ \begin{array}{rr} 2 & -1\\ 3 & -2 \\ -5 & 7 \\ \end{array} \right] \hspace{.5in} C= \left[ \begin{array}{rrr} 1 & 0 & -7\\ 4 & 6 & -2\\ -7 & 0 & 1 \\ \end{array} \right] \] \begin{enumerate} \item $2A-3B$ \item $AB$ \item $BA$ \item $BC$ \end{enumerate} \item Given the matrix $B$ below, find $B^{-1}$. \[ \left[ \begin{array}{rrr} 7 & 11 & 3 \\ 5 & 8 & 2 \\ 1 & 2 & 1 \end{array} \right] \] \item Are the two matrices below inverses? Explain why or why not. A simple yes or no (even if correct) is not worth anything. Hint: Try not to make this harder then it needs to be. \[ \left[ \begin{array}{rrr} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 2 & 2 & 1 \end{array} \right] \hspace{.5in} \left[ \begin{array}{rrr} -1/3 & 1 & -1/3 \\ 0 & -1 & 1 \\ 2/3 & 0 & -1/2 \end{array} \right] \] \item Given the fact that the two matrices below are inverses, solve the below system of equations \underline{using} inverse matrices. \[ A=\left[ \begin{array}{rrr} 2 & -1 & -2 \\ 3 & -1 & 1 \\ 1 & 1 & -1 \end{array} \right] \hspace{.5in} A^{-1}=\left[ \begin{array}{rrr} 0 & 1/4 & 1/4 \\ -1/3 & 0 & 2/3 \\ -1/3 & 1/4 & -1/12 \end{array} \right] \] \[ \begin{array}{rrrrrrr} 2x & - & y & - & 2z & = & 7 \\ 3x & - & y & + & z & = & 5 \\ x & + & y & - & z & = & -2 \\ \end{array} \] \item Graph the solution to the inequalities and find all the corners of the shaded region. \begin{eqnarray*} y - 2x & \geq & 0 \\ y + x & \geq & 3 \\ y & \leq & 10 \end{eqnarray*} \item You spend some time researching VW camper vans and studying their acceleration. You test the following model: The time $t$, in \emph{minutes}, that it takes a Volkswagen camper van to accelerate to $x$ miles per hour can be modeled by the equation: \[ t = .01(-597.62+.1976x^2 +21.017x) \] How fast can the camper van go after 11 minutes? (Your answer may not reflect the acceleration in your car, but remember that this is a VW.) \item You have a business and you model the number of employees you have by the equation \[ N=-.23t^2+4.89t+52.12 \] where $t$ is the number of years past $1999$ and $N$ is the number of employees. Do you have a maximum or minimum number of employees. When does this happen, and how many employees? (Write it in a good sentence!!!) Extra credit-2 points: When does your business die how do you know it dies then? \item \begin{enumerate} \item Write the following equation in logarithm form \[ 5^{-3}=\frac{1}{125} \] \item Write the following equation in exponential form \[ \log_3 \frac{1}{9}=-2 \] \end{enumerate} \item Get a decimal approximation of $\log_2 10$, and check your answer using exponentials. (Show your work.) \item The length $x$, in centimeters, of brook trout is modeled by the equation \[ x=60-50e^{-.05t} \] where $t$ is the age of the fish in months. You catch a 55~cm trophy brook trout. How old is it? (Answer in sentences!!) \item Volkswagen camper owners have found that the more stickers on their buses, the more likely they are to be pulled over by the police and illegally searched. It is somehow determined that if you drive your VW bus from coast to coast, with $x$ stickers on your bus, the chance $P$, in percentage, that you will be pulled over and illegally searched at least once is \[ P=90 (.12)^{.91^x} \] \begin{enumerate} \item What is your chance of being illegally searched with no stickers on your bus? (Sentences!!) \item How many stickers can you have on your bus if you are willing to accept a $30\%$ risk of being illegally searched. (I'll give you 1 point extra credit if you also have the exact answer, no decimals. In either case, you do need the decimal answer.) (Sentences!!) \end{enumerate} \end{enumerate} \newpage \hrule \vspace{.2in} Old tests from Fall 99: \begin{enumerate} \item The following function is used to determine a persons water bill: \[ C(x) = \left\{ \begin{array}{ll} 1.5x & \mbox{ if } 0 \leq x \leq 100 \\ 1.5x +1.2(x-100) & \mbox{ if } 100 < x \leq 1000 \\ 1000 + .7(x-1000) & \mbox{ if } x > 1000 \\ \end{array} \right. \] Where $C(x)$ is the cost in dollars for $x$ thousand gallons of water. \begin{enumerate} \item What is $C(250)$? \item Write a sentence explaining what $C(250)$ means in terms of the water bill. \end{enumerate} \item If $f(x)=\frac{x}{x^2-1}$ and $g(x)=x^3-1$, find, $f\circ g(x)$. (You don't need to simplify it). \item For the function $h$ below, find $f$ and $g$ so that $h(x)=f(g(x))$. \[ h(x)=\frac{52}{\sqrt{5x^{4}-6x}} \] \item For the function $f(x)=3x^2$, simplify the following: \[ \frac{f(x+h)-f(x)}{h} \] \item Lets say you are car shopping. You are dead set on a New Beetle. Let say that Volkswagen sells the Beetle to Dave Strong (a VW dealer) and 20\% of that price is VW's mark-up. Then, on top of that, 35\% of the sticker price is Dave Strong's mark-up. The Beetle costs VW \$9250. Answer the following questions: (As usual, I need to see the work for you to get credit.) \begin{enumerate} \item What does Dave Strong pay VW for the New Beetle? \item What is the sticker price of the Beetle? \end{enumerate} \item Graph the line $4x+5y=8$. Find the coordinates of the $x$-intercept and the $y$-intercept and label these points on your graph. \item Suppose that an appliance store has found that the demand for washing machines is 100 if the price is \$350. They have also found that the demand is 150 when the price is \$310. Assuming the demand function is linear, find the equation for demand (using $p$ for price and $q$ for quantity.) \item The supply and demand functions for a product are given below. Find the equilibrium point. \begin{eqnarray*} \mbox{Demand: } & p = 480-3q \\ \mbox{Supply: } & p = 17q+80 \end{eqnarray*} \item Suppose you are working at Gray Whale and sell a bunch of cd's. You sell some cd's at \$8 each and some at \$9 each. You end up selling 87 cd's and getting \$732. How many cd's do you sell at \$8 and how many at \$9? (You will \underline{not} get credit without setting up the equations.) \item You are selling belts for \$12 a belt. Every month your fixed costs are \$1600 and the belts cost you \$8 per belt. \begin{enumerate} \item Find the cost, revenue and profit functions. \item Find the break-even point. \end{enumerate} \item For the matrices below, find the following. \[ A=\left[ \begin{array}{rrrr} 1 & 2 & 5 & -1 \\ -4 & -3 & 23 & 0 \end{array} \right] \hspace{1in} B=\left[ \begin{array}{rr} -7 & -1 \\ 0 & 3 \\ 1 & -12 \\ 6 & 0 \end{array} \right] \] \begin{enumerate} \item Order of $A$. \item Order of $B$. \item $A+B$ \end{enumerate} \item Given the matrices \[ A=\left[ \begin{array}{rrr} 2 & 3 & 0 \\ 1 & 2 & 5 \end{array} \right] \hspace{1in} B=\left[ \begin{array}{rr} 0 & 2 \\ 1 & 4 \end{array} \right] \hspace{1in} C=\left[ \begin{array}{rr} -1 & 2 \\ 2 & 3 \end{array} \right] \] 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 $A-4C$ \end{enumerate} \item Given matrices $A$ and $A^{-1}$, and \emph{using these matrices}, solve the system of equations below. (You must set up the matrix equation to get full credit!) \[ A=\left[ \begin{array}{rrrr} 2 & 3 & 0 & 1 \\ 1 & 0 & 2 & 3 \\ 0 & 1 & 0 & 2 \\ 1 & 0 & 2 & 2 \end{array} \right] \hspace{1in} A^{-1}=\left[ \begin{array}{rrrr} 1/2 & 5/2 & -3/2 & -5/2 \\ 0 & -2 & 1 & 2 \\ -1/4 & -9/4 & 3/4 & 11/4 \\ 0 & 1 & 0 & -1 \end{array} \right] \] \[ \begin{array}{rrrrrrrrr} 2x & + & 3y & & & + & w & = & 2 \\ x & & & + & 2z & + & 3w & = & -2 \\ & & y & & & + & 2w & = & 4 \\ x & & & + & 2z & + & 2w & = & 8 \end{array} \] \item Find the inverse of the following matrix. You must show your work (calculator is not ok!) \[ \left[ \begin{array}{rrr} 0 & 2 & 1 \\ 3 & 0 & 1 \\ 1 & 1 & 1 \end{array} \right] \] \item You have \$60,000 to invest. Your choices are investing at 8\% per year and 14\% per year (a more risky investment). You want to earn \$5967 per year on interest. How much do you need to invest at each rate? (Use any method learned in class to solve this, just make sure you show your work!) \item Are the following matrices inverses of each other. Explain why or why not. \[ C=\left[ \begin{array}{rrrr} 0 & 2 & 2 & 1 \\ 0 & 2 & 3 & 1 \\ 3 & 0 & 1 & 3 \\ 1 & 0 & 0 & 3 \end{array} \right] \hspace{1in} D=\left[ \begin{array}{rrrr} 1/2 & -1 & 1/2 & -1/2 \\ 19/12 & -13/6 & 1 & 13 \\ -1 & 2 & 0 & -1 \\ -1/6 & 1/3 & -1/6 & 5/6 \end{array} \right] \] \item \begin{enumerate} \item Graph the solution to the inequality \[ -2x+5 > \frac{x-7}{2} \] \item Budget car rental rents a car for \$42 per day. Economy car rental rents the same car for \$25 per day, but charges an initial fee of \$85. Set up and solve an inequality that expresses the number of days that it is cheaper to rent from Economy. (So, you need to find the number of days that it would be cheaper to rent from Economy, but you need to set it up correctly too.) \end{enumerate} \item \begin{enumerate} \item Graph the system of inequalities \begin{eqnarray*} x+2y & \leq & 10 \\ 2x+y & \leq & 14 \\ x & \geq & 2 \\ y & \geq & 1 \end{eqnarray*} \item Find the maximum and minimum of $R=2x-3y$ (and where this maximum and minimum occurs) for $x$ and $y$ subject to the constraints above. \end{enumerate} \item You and a friend are making bumper stickers. You make 2 varieties: simple ones, and fancy ones. The simple sticker takes 15 minutes to make and the fancy ones take 30 minutes to make. You sell the simple stickers for \$2 each and the fancy ones for \$3 each. Between the two of you, you have 30 hours total per week to make stickers. The sticker market is such that you can only sell a total of 75 stickers each week. How many of each stickers should you make of each type to maximize your revenue? (Hint: Note the difference between hours and minutes. To get full credit you have to find the right inequalities, and graph the inequalities. Remember to tell me what your variables stand for!) \item Set up the simplex matrix used to solve the linear programming problem below. DO NOT SOLVE, JUST SET UP THE SIMPLEX MATRIX. (Task A) Maximize $f=3x+5y+11z$ subject to \begin{eqnarray*} 2x+3y+4z & \leq & 60 \\ x +4y+z & \leq & 48 \\ 5x+y+z & \leq & 50 \end{eqnarray*} \item A simplex matrix is given below. Manipulate the matrix (as discussed in class and in the book) until you can read off the solution. Assume that all variable are nonnegative. DO NOT READ OFF THE SOLUTION, JUST GET TO THE POINT WHERE YOU COULD. (Task B) \[ \left[ \begin{array}{rrrrr|r} 1 & 5 & 1 & 0 & 0 & 25 \\ 2 & 5 & 0 & 1 & 0 & 30 \\ \hline -2 & -6 & 0 & 0 & 1 & 0 \end{array} \right] \] \item I have taken a linear programming problem and turned it into a simplex matrix. This is the matrix on the left. I then manipulated this matrix to get the solution. This is the matrix on the right. (Task C) \[ \left[ \begin{array}{rrrrrrr|r} 2 & 7 & 9 & 1 & 0 & 0 & 0 & 100 \\ 6 & 5 & 1 & 0 & 1 & 0 & 0 & 145 \\ 1 & 2 & 7 & 0 & 0 & 1 & 0 & 90 \\ \hline -2 & -5 & -2 & 0 & 0 & 0 & 1 & 0 \end{array} \right] \longrightarrow \left[ \begin{array}{rrrrrrr|r} 0 & 1 & 13/8 & 3/16 & -1/16 & 0 & 0 & 155/16 \\ 1 & 0 & -19/16 & -5/32 & 7/32 & 0 & 0 & 515/32 \\ 0 & 0 & 79/16 & -7/32 & -3/32 & 1 & 0 & 1745/32 \\ \hline 0 & 0 & 15/4 & 5/8 & 1/8 & 0 & 1 & 645/8 \end{array} \right] \] Answer the following questions \begin{enumerate} \item How many variables did we start with and what are they? \item How many inequalities did we start with and what are they? \item What is the function to maximize? \item How many slack variables and what are they? \item What is the solution to the linear programming problem? (What is the maximum value for the function and what do all the variables need to be to get this max, including the slack variables.) \end{enumerate} \item You spend some time researching VW camper vans and studying their acceleration. You test the following model: The time $t$, in \emph{minutes}, that it takes a Volkswagen camper van to accelerate to $x$ miles per hour can be modeled by the equation: \[ t = .01(-597.62+.1976x^2 +21.017x) \] How fast can the camper van go after 11 minutes? (Your answer may not reflect the acceleration in your car, but remember that this is a VW.) \item Given cost and revenue functions, answer the questions. \begin{eqnarray*} C & = & 360+10x+.2x^2 \\ R & = & 50x-.2x^2 \end{eqnarray*} \begin{enumerate} \item Is there a maximum or minimum profit? \item What level of production gives the maximum or minimum profit? \item What is the maximum or minimum profit? \end{enumerate} \item The length $x$, in centimeters, of brook trout is modeled by the equation \[ x=66-58e^{-.04t} \] where $t$ is the age of the fish in months. You catch a 55~cm trophy brook trout. How old is it? (Answer in sentences!!) \item Volkswagen camper owners have found that the more stickers on their buses, the more likely they are to be pulled over by the police and illegally searched. It is somehow determined that if you drive your VW bus from coast to coast, with $x$ stickers on your bus, the chance $P$, in percentage, that you will be pulled over and illegally searched at least once is \[ P=88 (.11)^{.93^x} \] \begin{enumerate} \item What is your chance of being illegally searched with no stickers on your bus? (Sentences!!) \item How many stickers can you have on your bus if you are willing to accept a $40\%$ risk of being illegally searched. (Sentences!!) \end{enumerate} \item Get a decimal approximation of $\log_{17} 1205$, and check your answer using exponentials. (Show your work.) \item Suppose that you have \$5000 and you find somewhere to invest it at 19\% annual simple interest. How long do you need to invest your money in order to have \$7500? \item You have \$5000 and you invest it for 10 years at 6\% annual interest. What is the future value if the interest is \begin{enumerate} \item compounded annually? \item compounded monthly? \item compounded continuously? \end{enumerate} \item If you compound your 6\% annual interest continuously, and you start with \$5000, how long until you have \$8000? \item What is better and why? \begin{enumerate} \item 13.3\% annual interest compounded biannually, \item 13\% annual interest compounded continuously, \item 13.2\% annual interest compounded 6 times a year. \end{enumerate} \item Suppose that when you are 30 you get an inheritance of \$50,000. You decide that you will shove this money in a bank (with 7\% annual interest compounded monthly) for 30 years and then use it for your retirement. How much can you withdraw every month from the time you are 60 until you are 80? (You withdraw an equal amount each month) \item At the birth of your new baby, you start putting some money into an account at the beginning of every month. You account pays 8\% annual interest compounded monthly. How much to you need to deposit every month in order to have \$25,000 when you child is 18 and ready to start college? \item Suppose that you just started college. Rather then pay for your expenses as they came up, your parents dropped a lump sum of money into an account that pays 5\% annual interest compounded 6 times a year. You then receive \$800 every 2 months for 4 years. How much money did your parents have to drop in the account to make this possible? \item (EXTRA CREDIT - 6 points) \textbf{Important:} You will only get credit if you use the methods taught in class. Also, decimals are not OK. To get credit, leave your answer in a fraction. These points (if any) will be added to this test. \begin{enumerate} \item Find the sum of the first 8 terms of the geometric sequence below. \[ \frac{2}{3},3, \ldots \] \item You have an arithmetic sequence with 7th term 12 and 82nd term 387. What is the 193rd term? \end{enumerate} \end{enumerate} \end{document}