The following table gives the population breakdown (in thousands) of the U.S. population in 2008 based on marital status and race or ethnic origin.

$$\begin{array}{lcccc} & & & & \text { Asian or } \\ & \begin{array}{c} \text { White } \\ (W) \end{array} & \begin{array}{c} \text { Black } \\ (\boldsymbol{B}) \end{array} & \begin{array}{c} \text { Hispanic } \\ (\boldsymbol{H}) \end{array} & \begin{array}{c} \text { Pacific } \\ \text { Islander }(\boldsymbol{A}) \end{array} \\ \hline \text {Never married }(N) & 54,205 & 13,547 & 12,021 & 3518 \\ \hline \text {Married }(M) & 106,517 & 9577 & 16,111 & 6741 \\ \hline \text {Widowed }(I) & 11,968 & 1740 & 1068 & 507 \\ \hline \begin{array}{l} \text {Divorced} / \\ \text {separated }(D) \end{array} & 23,046 & 4590 & 3477 & 665 \\ \hline \end{array}$$

$(M \cup I) \cap A$

Use the mgf to show that if X follows an exponential distribution, $c X(c>0)$ does also.

Give an example of a $3 \times 3$ matrix A with nonzero integer entries such that 1, 2, and 3 are the eigenvalues of A.

Let $X_{1}$ and $X_{2}$ be independent standard normal random variables. Show that the

Find the Fourier series for the indicated function on the indicated interval. Plot the function and two partial sums of your choice over the interval.

$$f(x)=\left\{\begin{array}{ll} 1+x, & \mathrm { for } \ -1 \leq x \leq 0, \\ 1-x, & \mathrm { for } \ 0<x \leq 1, \end{array}\right.$$

on [-1, 1]

After rounding the final turn in the bell lap, two runners emerge ahead of the pack. When Runner A. is 200 ft from the finish line, his speed is 22 ft/sec, a speed that he maintains until he crosses the line. At that instant of time, Runner B, who is 20 ft behind Runner A and running at a speed of 20 ft/sec, begins to sprint. Assuming that Runner B sprints with a constant acceleration, what minimum acceleration will enable him to cross the finish line ahead of Runner A?

Show that the given signal is a solution of the difference equation. Then find the general solution of the difference equation. $y_{k}=k-2 ; y_{k+2}-4 y_{k}=8-3 k$

A drug is carried into an organ of volume V $cm^3$ by a liquid that enters the organ at the rate of a $\mathrm{cm}^{3} / \mathrm{sec}$ and leaves it at the rate of b $\mathrm{cm}^{3} / \mathrm{sec}$. The concentration of the drug in the liquid entering the organ is c $\mathrm{g} / \mathrm{cm}^{3}$. If the concentration of the drug in the organ at time t is increasing at the rate of $x^{\prime}(t)=\frac{1}{V}\left(a c-b x_{0}\right) e^{-b t v}$ $\mathrm{g} / \mathrm{cm}^{3} / \mathrm{sec}$ and the concentration of the drug in the organ initially is $x_{0} \mathrm{g} / \mathrm{cm}^{3}$, show that the concentration of the drug in the organ at time t is given by $x(t)=\frac{a c}{b}+\left(x_{0}-\frac{a c}{b}\right) e^{-b c v}$

Find the derivative of the function. $f(x)=\frac{\ln x}{x+1}$

Use the table, which shows the years of experience of 14 registered nurses and their annual salaries (in thousands of dollars).

$$\begin{matrix} \text{Years of} & \text{0.5} & \text{2} & \text{4} & \text{5} & \text{7} & \text{9} & \text{10}\\ \text{experience, x} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Annual salary (in} & \text{45.2} & \text{49.9} & \text{54.7} & \text{59.3} & \text{61.4} & \text{62.9} & \text{66.0}\\ \text{thousands of dollars), y} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Years of} & \text{12.5} & \text{13} & \text{16} & \text{18} & \text{20} & \text{22} & \text{25}\\ \text{experience, x} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Annual salary (in} & \text{67.1} & \text{65.3} & \text{68.4} & \text{70.6} & \text{69.5} & \text{73.9} & \text{71.6}\\ \text{thousands of dollars), y} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \end{matrix}$$

A salary analyst claims that the population has a significant correlation for $\alpha=0.01$. Test this claim.

For the following exercise, evaluate the function. Give the exact value. $\sin ^{-1}\left(\sin \left(\frac{\pi}{3}\right)\right)$

Find the indefinite integral. $\int\left(1+u+u^{2}\right) d u$

Find the indefinite integral. $\int\left(1+x+e^{x}\right) d x$

If a $4 \times 7$ matrix A has rank 3, find dim Nul A, dim Row A, and rank $A^{T}$.