In given problem, find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=e^x \cos y \mathbf{i}+e^x \sin y \mathbf{j}+z \mathbf{k}$$

Ithows three residual plots and a normal probability plot of residuals. For each part, decide whether the graph suggests a violation of one or more of the assumptions for regression inferences. Explain your answers.

Refer to the Lichen Radionuclide Baseline Research project. University of Alaska researchers determined the amount of the radioactive element cesium-137 for nine lichen specimens sampled from various Alaskan locations. The data values (measured in microcuries per milliliter) are given in the table.

$$\begin{array}{lllll} \hline 0.0040868 & 0.0157644 & 0.0023579 & 0.0067379 & \\ 0.0165727 & 0.0067379 & 0.0078284 & 0.0111090 & 0.0100518 \\ \hline \end{array}$$

a. Describe the population of interest to the researchers.

b. Estimate the mean of the population using a $95 \%$ confidence interval.

c. Make an inference about the magnitude of the population mean.

d. For the inference to be valid, how must the population data be distributed?

Some anatomy students were saying that a man's bulbo-urethral glands (and the urethral glands) act like city workers who come around before a parade and clear the street of parked cars. What did they mean by this analogy?

A multiple regression analysis of 57 cases was performed with four explanatory variables. Suppose that SSM=16.5 and SSE=100.8.

(a) Find the value of the F statistic for testing the null hypothesis that the coefficients of all the explanatory variables are zero.

(b) What are the degrees of freedom for this statistic?

(c) Find bounds on the P-value using Table E. Show your work.

(d) What proportion of the variation in the response variable is explained by the explanatory variables?

A healthy cereal should be low in both calories and sodium. Data for 77 cereals were examined and judged acceptable for inference. The 77 cereals had between 50 and 160 calories per serving and between 0 and 320 mg of sodium per serving. Here's the regression analysis:

$$\newline$$

Dependent variable is : Sodium$\newline$ R-squared$= 9.0\%$ $\newline$ $s = 80.49$ with $77- 2 = 75$ degree of freedom $\newline$

$$\begin{alignat*}{7} \textbf{Variable} \quad &\textbf{Coefficient} \quad &\textbf{SE(Coeff)} \quad &\textbf{t-Rate} \quad &\textbf{P-Value} \\ \text{Intercept} \quad & 21.4143\quad & 51.47\quad & 0.416\quad & 0.6786\\ \text{Verbal} \quad & 1.29357\quad & 0.4738\quad & 2.73\quad & 0.0079 \end{alignat*}$$

Is there an association between the number of calories and the sodium content of cereals? Explain.

Using DeMorgan’s Law, write an expression for the complement of F if $F(x,y,z) = (x\prime + y)(x + z)(y\prime + z)\prime$.

On January 1, 2012, Tropical Paradise borrows $40,000 by agreeing to a 6%, fiveyear note with the bank. The funds will be used to purchase a new BMW convertible for use in promoting resort properties to potential customers. Loan payments of$773.31 are due at the end of each month with the first installment due on January 31, 2012.

Required: Record the issuance of the installment note payable and the first two monthly payments.

For each of the three crossed aldol reactions described, indicate which, if any, of the multiple possible products should predominate in the reaction mixture and explain why.

Prepare a multiple-step income statement for 2016 applying generally accepted accounting principles. The income tax rate is 40%

To check your equations of tangent lines, first graph f(x) = 3/x. Select Tangent from the DRAW menu, and enter first x = -2 and then $x=-\frac{1}{2}$.

n article discusses the market for autographs by Mickey Mantle, the superstar centerfielder for the New York Yankees during the 1950s and 1960s: "At card shows, golf outings, charity dinners, Mr. Mantle signed his name over and over." One expert on sport autographs is quoted as saying, "He was a real good signer.... He is not rare." Yet the article quotes another expert as saying, "Mr. Mantle's autograph ranks No. 3 of most-popular autographs, behind Babe Ruth and Muhammad Ali." A baseball signed by Mantle is likely to sell for the relatively high price of $\$ 250$ to $\$ 400$. By contrast, baseballs signed by Whitey Ford, a teammate of Mantle's on the Yankees, typically sell for less than $\$ 150$. Use one graph to show both the demand and supply for autographs by Whitey Ford and the demand and supply for autographs by Mickey Mantle. Show how it is possible for the price of Mantle's autographs to be higher than the price of Ford's autographs, even though the supply of Mantle autographs is larger than the supply of Ford autographs. Source: Beth DeCarbo, "Mantle Autographs Not Rare, but Collectors Don't Care," Wail Street Jourmal, August 4, 2008.

Let $a$ and $b$ be positive integers. Prove that if $d=(a, b), a=a_0 d$, and $b=b_0 d$, then the least common multiple of $a$ and $b$ is $a_0 b_0 d$.

Use an appropriate graphing device to plot accurately several trajectories in the phase plane and the corresponding graphs of $x_1$ versus $t$ and $x_2$ versus $t$.

$$x^{\prime}=\left(\begin{array}{ll}3 & -2 \\ 4 & -1\end{array}\right) x$$