Exam 7: Analytic Geometry and Nonlinear Systems

Find the partial fraction decomposition for the rational expression $\frac{x-11}{x^{2}-2 x-3}$ .

During one period, the average price of a gallon of gasoline in a town changed from day to day. The table shows the average price, in dollars, for each of several consecutive days, where 0 represents Sunday, 1 represents Monday, and so on. Use the data points $(0,3.50),(2,3.15)$ , and $(4,3.6)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data.

Perform the following matrix operations if possible. (a) $\left(\begin{array}{lll}1 & -7 & 3\end{array}\right)-\left(\begin{array}{ll}4 & -5\end{array}\right)$ (b) $\left[\begin{array}{rrr}1 & 6 & 7 \\ 0 & -3 & 5\end{array}\right]-2\left[\begin{array}{rrr}6 & -5 & 2 \\ -1 & 0 & 3\end{array}\right]$ (c) $\left(\begin{array}{lll}4 & 3 & -2\end{array}\right)\left[\begin{array}{rr}1 & -2 \\ -1 & 5 \\ 6 & 0\end{array}\right]$

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}6 & 3 \\ -8 & 2\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}3 & 1 & -2 \\ 1 & -4 & 3 \\ 7 & 11 & -12\end{array}\right]$

Consider the system of equations $4 x^{2}+y^{2}=20$ $-2 x+3 y=10$ (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

The solution set of a system of inequalities is shown below. Which of the following systems is it? (a) $y-2 x \leq-9$ $y \leq-x^{2}+6 x-4$ (b) $y-2 x \leq-9$ $y \geq-x^{2}+6 x-4$ (c) $y-2 x \geq-9$ $y \leq-x^{2}+6 x-4$ (d) $y-2 x \geq-9$ $y \geq-x^{2}+6 x-4$

Consider the system of equations $3 x^{2}+y^{2}=12$ $2 x+2 y=-4$ (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

Consider the system of equations x+4y-2z=0 -2x-6y+5z=5 3x+14y-4z=8 (a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system. (b) Find $A^{-1}$ . (c) Use the matrix inverse method to solve the system. (d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

A small city conducts a population census every two years; the census was begun in the year 2000. The population declined in the early 2000s but began to grow in later years. The table shows the population, in hundred thousands, for several years with 2000 represented by year 0,2002 by year 2 , and so on. Use the data points $(0,2.21),(4,1.95)$ , and $(8,2.30)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data.

A restaurant supply company must ship at least 80 cases of seafood to two midwest restaurants. Each restaurant can accommodate at most 80 cases of seafood. Restaurant A already has 20 cases of seafood on hand, while restaurant B already has 16 cases. It costs $\$ 9.60$ to ship a case of seafood to restaurant A and $\$ 8$ to ship a case of seafood to restaurant B. How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Suppose that $A$ is a $n \times 2$ matrix and $B$ is a $2 \times n$ matrix. (a) Can $A B$ be found? If so, give the dimensions of the resulting matrix. (b) Can $B A$ be found? If so, give the dimensions of the resulting matrix. (c) Does $A B=B A$ ? Explain why or why not. (d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

Solve the following system using one of the row echelon methods. x+y+2z =4 3x-2y+4z =12 2x-2y+z =1

Solve the following system using one of the row echelon methods. x+5y+z =-7 4x+2y+7z =32 2x-3y-z =19

Find the partial fraction decomposition for the rational expression $\frac{5 x+11}{x^{2}+5 x+6}$ .

Consider the system of equations x+3y+z =3 2x+7y =16 -3x-7y-6z =7 (a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system. (b) Find $A^{-1}$ . (c) Use the matrix inverse method to solve the system. (d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

Find the partial fraction decomposition for the rational expression $\frac{7 x+1}{x^{2}-x-20}$ .

A restaurant supply company must ship at least 60 cases of lettuce to two midwest restaurants. Each restaurant can accommodate at most 60 cases of lettuce. Restaurant A already has 15 cases of lettuce on hand, while restaurant B already has 12 cases. It costs $\$ 7.20$ to ship a case of lettuce to restaurant A and $\$ 6$ to ship a case of lettuce to restaurant B. How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Find the partial fraction decomposition for the rational expression $\frac{5 x^{2}+30 x+56}{(x-4)(x+4)^{2}}$ .

A restaurant supply company must ship at least 40 cases of vegetables to two midwest restaurants. Each restaurant can accommodate at most 40 cases of vegetables. Restaurant A already has 10 cases of vegetables on hand, while restaurant $\mathrm{B}$ already has 8 cases. It costs $\$ 4.80$ to ship a case of vegetables to restaurant $\mathrm{A}$ and $\$ 4$ to ship a case of vegetables to restaurant $B$ . How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Solve the following system using one of the row echelon methods. x-3y+2z=1 2x-4y+3z=3 3x-7y+6z=3