(i) sketch the differential equation in the $y-y^{\prime}$ plane by hand. (ii) Draw the phase line diagram, clearly stating the location and stability of the equilibria. (iii) State the long-term behavior for all initial conditions. (iv) Assuming y = y(x), sketch the corresponding graph in the traditional x-y plane. $y^{\prime}=-y^{2}$

Verify that the given function is a solution to the differential equation by substituting it into the differential equation and showing that the equation holds true. Assume the interval is $(-\infty, \infty)$ unless otherwise stated. Do NOT attempt to solve the differential equation. $y=2, \frac{d y}{d x}=x^{3}(y-2)^{2}$

Find either F(s) or f(t), as indicated. $\mathscr{L}\left\{\left(1-e^{t}+3 e^{-4 t}\right) \cos 5 t\right\}$

Convert the system to an augmented matrix and then find all solutions by reducing the system to echelon form and back substituting.

$$\begin{array}{r} 2 x_{1}+6 x_{2}-9 x_{3}-4 x_{4}=0 \\ -3 x_{1}-11 x_{2}+9 x_{3}-x_{4}=0 \\ x_{1}+4 x_{2}-2 x_{3}+x_{4}=0 \end{array}$$

During inspiration, intrapulmonary pressure is a. greater than atmospheric pressure b. less than atmospheric pressure c. greater than intrapleural pressure d. less than intrapleural pressure

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards. a. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time. b. What effect does the low response rate have on the reliability of the sample? c. Are these problems examples of sampling error or nonsampling error? d. During the same year, George Gallup conducted his own poll of 30,000 prospective voters. His researchers used a method they called "quota sampling" to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?

Use the technology of your choice to a. obtain and interpret the standard error of the estimate. b. obtain a residual plot and a normal probability plot of the residuals. c. decide whether you can reasonably consider Assumptions 1-3 for regression inferences met by the two variables under consideration. From the Statistical Abstract of the United States, we obtained data on percentage of gross domestic product (GDP) spent on health care and life expectancy, in years, for selected countries. Those data are provided on the WeissStats CD. Do the required parts separately for each gender.

Please provide the following information. (a) What is level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? (c) Find the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha$? (e) State your conclusion in the context of the application? Note: Answers may vary due to rounding. Would you favor spending more federal tax money on the arts? This question was asked by a research group on behalf of The National Institute. Of a random sample of $n_{1}=93$ politically conservative voters, $r_{1}=21$ responded yes. Another random sample of $n_{2}=83$ politically moderate voters showed that $r_{2}=22$ responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use $\alpha=0.05$.

Show that the Taylor series for $e^{x}, \sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ converges to $e^x$ by showing that the error for the nth Taylor approximation approaches zero as follows: a. Use the Ratio Test to show that the series $\sum_{n=0}^{\infty} \frac{|x|^{n}}{n !}$ converges for all x. b. Use the nth term test to conclude that $\frac{|x|^{n}}{n !} \rightarrow 0$ as $n \rightarrow \infty$ c. Use part (b) to show that $\left|R_{n}(x)\right| \leq \frac{e^{|x|}}{(n+1) !}|x|^{n+1} \rightarrow 0$ as $n \rightarrow \infty$ for any fixed value of x.

Anthropologists are still trying to unravel the mystery of the origins of the Etruscan empire, a highly advanced Italic civilization formed around the eighth-century B.C. in central Italy. Were they native to the Italian peninsula or, as many aspects of their civilization suggest, did they migrate from the East by land of their civilization suggest, did they migrate from the East by land or sea? The maximum head breadth, in millimetres, of 70 modern Italian male skulls and that of 84 preserved Etruscan male skulls Italian male skulls and that of 84 preserved Etruscan male skulls were analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats site. Use the technology of your choice to solve parts (a)-(c). a. Perform a two~standard~deviations F-test at the 5% significance level to decide whether the data provide sufficient evidence to conclude that variations in skull measurements differ between the two populations. b. Use the two-standard-deviations F-interval procedure to determine a 95% confidence interval for the ratio of the standard deviations of skull measurements of the two populations. c. Obtain a normal probability plot for each sample. d. In light of your plots in panic), does conducting the inferences you did in parts (a) and (b) seem reasonable? Explain your answer.

For each initial value problem: a. Use an Euler’s method graphing calculator program to find the estimate for y(2). Use the interval [0, 2] with n = 50 segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at x = 2. Compare this actual value of y(2) with the estimate of y(2) that you found in part (a).

$$\begin{aligned} &y^{\prime}=\frac{x}{y}\\ &y(0)=1 \end{aligned}$$

Solve the system by the method of substitution.

$$\left\{\begin{array}{rr} 3 x-2 y= & 3 \\ -6 x+4 y= & -6 \end{array}\right.$$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says “and check,” verify that your answer is a solution. $y^{\prime}=\frac{y^{2}}{x^{2}}$ and check

Use the definition of the derivative to show that the derivative of $f(t)=\cos t$ is $f^{\prime}(t)=-\sin t$.