A study of driving costs based on 2012 medium-sized sedans found that the average cost (car payments, gas, insurance, upkeep, and depreciation) in cents per mile is approximately $C(x)=\frac{1910.5}{x^{1.72}}+42.9 \ (5 \leq x \leq 20)$ where x (in thousands) denotes the number of miles the car is driven each year. Show that the graph of C is concave upward on the interval (5, 20). What does your result tell you about the average cost of driving a 2012 medium-sized sedan in terms of the number of miles driven?

Order the integers in each set from least to greatest. $\{0,-23,75,-15,24\}$

Suppose that a bivariate normal distribution has $\mu_{X}=\mu_{Y}=0 \text { and } \sigma_{X}=\sigma_{Y}=1$. Sketch the contours of the density and the lines $E(Y | X=x)$ and $E(X | Y=y)$ for $\rho=0$, .5, and .9.

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.

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

$(M \cup I) \cap A$

Let U and V be independent random variables with means $\mu$ and variances $\sigma^{2}$. Let $Z=\alpha U+V \sqrt{1-\alpha^{2}}$. Find E(Z) and $\rho_{U Z}$.

Suppose that a radioactive substance decays according to the model $N=N_{0} e^{-\lambda t}$. (a) Show that affer a period of $T_{\lambda}=1 / \lambda$, the material has decreased to $e^{-1}$ of its original value. $T_{\lambda}$ is called the time constant and it is defined by this property. (b) A certain radioactive substance has a half-life of 12 hours. Compute the time constant for this substance. (c) If there are originally 1000 mg of this radioactive substance present, plot the amount of substance remaining over four time periods $T_{\lambda}$.

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?

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 $x(t)=\frac{a c}{b}+\left(x_{0}-\frac{a c}{b}\right) e^{-b c v}$

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

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

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

Use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years.

$$\begin{matrix} \text{Librarians, x} & \text{Library science teachers, y}\\ \text{49.1} & \text{56.6}\\ \text{50.9} & \text{57.6}\\ \text{52.9} & \text{59.7}\\ \text{54.7} & \text{61.6}\\ \text{55.7} & \text{64.3}\\ \text{56.4} & \text{67.0}\\ \text{57.0} & \text{70.0}\\ \text{57.2} & \text{70.8}\\ \text{57.6} & \text{73.3}\\ \text{58.1} & \text{72.4}\\ \text{58.9} & \text{73.0}\\ \text{59.9} & \text{72.3}\\ \end{matrix}$$

Construct a scatter plot for the data. Do the data appear to have a positive linear correlation, a negative linear correlation, or no linear correlation? Explain.

Find the density function of $Y=e^{Z}$, where $Z \sim N\left(\mu, \sigma^{2}\right)$. This is called the lognormal density, since log Y is normally distributed.

In the children's game rock, paper, scissors, two players simultaneously extend one hand in the form of a fist (rock), a flat palm (paper), or two fingers extended (scissors). If they both make the same choice, the game is a tie. Otherwise, rock beats scissors (because rock can crush scissors), scissors beats paper (because scissors can cut paper), and paper beats rock (because paper can cover rock). Whoever wins gains one point; in the case of a tie, no one gains any points. (a) Find the optimum strategy for each player and the value of the game. (b) In hindsight, how could you have guessed the answer to part (a) without doing any work?