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?

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.

Show that a group of order 24 with no element of order 6 is isomorphic to $S_{4}$.

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?

Suppose that the probability of living to be older than 70 is .6 and the probability of living to be older than 80 is .2. If a person reaches her 70th birthday, what is the probability that she will celebrate her 80th?

The number of saltwater crocodiles in a certain area of northern Australia is given by $P(t)=\frac{300 e^{-0.024 t}}{5 e^{-0.024 t}+1}$ a. How many crocodiles were in the population initially? b. Show that $\lim _{t \rightarrow \infty} P(t)=0$. c. Plot the graph of P in the viewing window $[0,200] \times [0,70]$.

Use logarithmic differentiation to find the derivative of the function. $y=\left(x^{2}+1\right)^{x}$

Prove the following two identities both algebraically and by interpreting their meaning combinatorially. a.

$$\left(\begin{array}{l} n \\ r \end{array}\right)=\left(\begin{array}{l} n \\ n-r \end{array}\right)$$

b.

$$\left(\begin{array}{l} n \\ r \end{array}\right)=\left(\begin{array}{l} n-1 \\ r-1 \end{array}\right)+\left(\begin{array}{c} n-1 \\ r \end{array}\right)$$

Find an invertible matrix P and a matrix C of the form

$$\left[\begin{array}{rr} a & -b \\ b & a \end{array}\right]$$

such that the given matrix has the form $A=P C P^{-1}$.

$$\left[\begin{array}{rr} 4 & -2 \\ 1 & 6 \end{array}\right]$$

Find the upper and lower central series for $A_{4}$ and $S_{4}$.

A sample of sulfur weighing 0.210 g was dissolved in 17.8 g of carbon disulfide,$\mathrm{CS}_{2}\left(\mathrm{K}_{\mathrm{b}}=2.43^{\circ} \mathrm{C} / \mathrm{m}\right)$. If the boiling point elevation was 0.107 $^{\circ} \mathrm{C}$, what is the formula of a sulfur molecule in carbon disulfide?

Find the absolute maximum value and the absolute minimum value, if any, of the function. $h(t)=8 t-\frac{1}{t^{2}}$ on [1, 3]