Evaluate each of the following definite integrals by thinking of the graphs of the functions, without any calculation. $\int_{2.345}^{2.345+1000 \pi} \cos t d t$

The World Series in baseball is won by the first team to win four games (ignoring the 1903 and 1919-1921 World Series, when it was a best of nine). Thus it takes at least four games and no more than seven games to establish a winner. As found on the Major League Baseball Web site in World Series Overview, historically, the lengths of the World Series are as given in the following table.

$$\begin{array}{c|c|c} \hline \begin{array}{c} \text { Number } \\ \text { of games } \end{array} & \text { Frequency } & \begin{array}{c} \text { Relative } \\ \text { frequency } \end{array} \\ \hline 4 & 20 & 0.20 \\ 5 & 23 & 0.23 \\ 6 & 22 & 0.22 \\ 7 & 35 & 0.35 \\ \hline \end{array}$$

a. If X denotes the number of games that it takes to complete a World Series, identify the possible values of the random variable X. b. Do the first and third columns of the table provide a probability distribution for X? Explain your answer. c. Historically, what is the most likely number of games it takes to complete a series? d. Historically, for a randomly chosen series, what is the probability that it ends in five games? e. Historically, for a randomly chosen series, what is the probability that it ends in five or more games? f. The data in the table exhibit a statistical oddity. If the two teams in a series are evenly matched and one team is ahead three games to two, either team has the same chance of winning game number six. Thus there should be about an equal number of six- and seven-game series. If the teams are not evenly matched, the series should tend to be shorter, ending in six or fewer games, not seven games. Can you explain why the series tend to last longer than expected?

Find the derivative of each function. $f(x)=\frac{\ln x}{x^{3}}$

In a recent Pew Research poll, 287 out of 522 randomly selected men in the United States were able to identify Egypt when it was highlighted on a map of the Middle East. When 520 randomly selected women were asked, 233 were able to do so. Construct and interpret a 95% confidence interval for the difference in the proportion of U.S. men and the proportion of U.S. women who can identify Egypt on a map.

_____ is a condition in which a person voids urine involuntarily.

Enter the formula for windchill index from the Graphing Calculator Exploration, but with y ( temperature) replaced by several temperatures: 20, 30, 40, and 50 degrees (using $y_1, y_2, y_3,$ and $y_4$). a. Graph the functions on the window [0, 45] by [-10, 60]. b. Notice that the lower temperature curves slope downward more steeply than the others. What does this mean about the effect of wind on a colder day? c. Use NDERIV or dy/dx to find the slope of the lowest and the highest curves at x = 10. Interpret the answers. d. Do your answers to part (c) support your conclusion in part (b)?

From 2008 to 2011, new car loan interest rates at auto finance companies were approximately $I(x)=0.106\left(x^{2}-6.85 x+14\right)\left(x^{2}-14.5 x+56\right)$ percent where x is the number of years after 2005. Differentiating using the Product Rule, find $I^{\prime}(3)$ and $I^{\prime}(5)$ and interpret your answers.

Anyone who has been outdoors on a summer evening has probably heard crickets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This phenomenon was studied in detail by George W. Pierce, a physics professor at Harvard. In the following, data, x is a random variable representing chirps per second and y is a random variable representing temperature $\left(^{\circ} \mathrm{F}\right)$. These data are on the statSpace CD-ROM.

$$\begin{array}{c|cccccccc} \hline x & 20.0 & 16.0 & 19.8 & 18.4 & 17.1 & 15.5 & 14.7 & 17.1 \\ \hline y & 88.6 & 71.6 & 93.3 & 84.3 & 80.6 & 75.2 & 69.7 & 82.0 \\ \hline \end{array}$$

$$\begin{array}{c|ccccccc} \hline x & 15.4 & 16.2 & 15.0 & 17.2 & 16.0 & 17.0 & 14.4 \\ \hline y & 69.4 & 83.3 & 79.6 & 82.6 & 80.6 & 83.5 & 76.3 \\ \hline \end{array}$$

Complete parts (a) through (e), given $\Sigma x=249.8, \Sigma y=1200.6, \Sigma x^{2}=4200.56 \Sigma y^{2}=96,725.86, \Sigma x y=20,127.47, \text { and } r=0.835$ .

Explain what it means if there is a negative association between the amount of fuel oil that a furnace uses and the outside temperature.

Susan is taking western civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 77% of the students in western civilization each term. Let n = 1, 2, 3, $\ldots$ represent the number of times a student takes western civilization until the first passing grade is received. (Assume the trials are independent.) (e) What is the expected number of attempts at western civilization Susan must make to have her (first) pass?

Susan is taking western civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 77% of the students in western civilization each term. Let n = 1, 2, 3, $\ldots$ represent the number of times a student takes western civilization until the first passing grade is received. (Assume the trials are independent.) What is the probability that Susan needs three or more tries to pass western civilization?

In the D-test we did not discuss what happens if D > 0 and $f_{x x}=0$. Why not? [Hint: Can this case occur?]

A sociologist is interested in the relation between x=number of job changes and y=annual salary (in thousands of dollars) for people living in the Nashville area. A random sample of 10 people employed in Nashville provided the following information:

$$\begin{matrix}\text{x (Number of job changes)} & \text{4} & \text{7} & \text{5} & \text{6} & \text{1} & \text{5} & \text{9} & \text{10} & \text{10} & \text{3}\\ \text{y (Salary in \$1000 ) } & \text{33} & \text{37} & \text{34} & \text{32} & \text{32} & \text{38} & \text{43} & \text{37} & \text{40} & \text{33}\\ \end{matrix}$$

$\Sigma x=60$; $\Sigma y=359$; $\Sigma x^{2}=442$; $\Sigma y^{2}=13,013$; $\Sigma x y=2231$ Verify that $S_{e} \approx 2.56$.

Many products come with owner registration or warranty cards. Usually, the consumer is asked a few questions about his or her family and household income. Random samples of warranty or registration cards fo