The concentration of hydrogen ions in a substance is denoted by $\left[\mathrm{H}^{+}\right]$, measured in moles per liter. The pH of a substance is defined by the logarithmic function $\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]$. This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base. a. Find the pH of the following substances. Round answers to one digit. b. Determine whether the substance is an acid or a base. i. $\operatorname{Eggs}:\left[\mathrm{H}^{+}\right]=1.6 \times 10^{-8} \mathrm{mol} / \mathrm{L}$ ii. Beer: $\left[\mathrm{H}^{+}\right]=3.16 \times 10^{-3} \mathrm{mol} / \mathrm{L}$ iii. Tomato Juice: $\left[\mathrm{H}^{+}\right]=7.94 \times 10^{-5} \mathrm{mol} / \mathrm{L}$

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

Use the data to (a) find the coefficient of determination $r^2$ and interpret the result, and (b) find the standard error of estimate $s_e$, and interpret the result. The U.S. voting age populations (in millions) and the number of ballots cast (in millions) for the highest office in federal elections for nine nonpresidential election years are shown in the table. The equation of the regression line is $\hat{y}=0.276 x+18.881$

$$\begin{matrix} \text{Voting age population, x} & \text{166.0} & \text{177.9} & \text{186.2} & \text{195.3} & \text{205.3}\\ \text{Ballots cast in federal} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{elections, y} & \text{67.6} & \text{65.0} & \text{67.9} & \text{75.1} & \text{72.5}\\ \text{Voting age population, x} & \text{215.5} & \text{225.5} & \text{236.0} & \text{245.7} & \text{ }\\ \text{Ballots cast in federal} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{elections, y} & \text{78.4} & \text{83.8} & \text{89.1} & \text{81.7} & \text{ }\\ \end{matrix}$$

Find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. $P(z < -1.22$ or $z > 1.32)$

The mass of a radioactive isotope at time t (in years) is $M(t)=200 e^{-0.14 t} \mathrm{g}$. What is the mass of the isotope initially? How fast is the mass of the isotope changing 2 years later?

For the following exercise, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.

$$g(x)=\left\{\begin{array}{l} \frac{3}{x-2}, x \neq 2 \\ 4, x=2 \end{array} ; g(0) ; g(-4) ; g(2)\right.$$

Complete the chart. Review each word, its root, and the mentor sentences. Use the information and your own knowledge to predict the meaning of each word. For each word, list at least two related words. Refer to a dictionary or other resources if needed.

$$\begin{array}{|l|l|l|l|} \hline \text { WORD } & \text { MENTOR SENTENCES } & \text { PREDICT MEANING } & \text { RELATED WORDS } \\ \hline \text { depiction} & \text { 1. The new president's vision for America is a } & & \\ \text {ROOT} & \text {depiction of peace, equality, and prosperity. } & & \\ \text {-pict -} & \text {2. Your proposal for the new playground should include some} & & \\ \text { "paint"} & \text { sort of depiction, such as a drawing or map.} & & \\ \hline \end{array}$$

Let m, g, and L be positive constants. (a) Show that $V(\theta, \omega)=\frac{1}{2} m L^{2} \omega^{2}+m g L(1-\cos \theta)$ is positive definite in an open neighborhood containing $(\theta, \omega)=(0,0)$. (b) Show that $\dot{V}(\theta, \omega)$ is negative semidefinite on solution trajectories of

$$\begin{array}{l} \dot{\theta}=\omega \\ \dot{\omega}=-\frac{\mu}{m} \omega-\frac{g}{L} \sin \theta \end{array}$$

on an open neighborhood containing $(\theta, \omega)=(0,0)$.

Let f be a constant function-that is, let f(x)=c, where c is some real number. Show that every number a gives rise to an absolute maximum and, at the same time, an absolute minimum of f.

Sample annual salaries (in thousands of dollars) for employees at a company are listed.

$$\begin{matrix} \text{42} & \text{36} & \text{48} & \text{51} & \text{39} & \text{39} & \text{42}\\ \text{36} & \text{48} & \text{33} & \text{39} & \text{42} & \text{45} & \text{50}\\ \end{matrix}$$

Find the sample mean and the sample standard deviation.

Prove that $G^{i}$ is a characteristic subgroup of G for all i.

(a) identify the claim and state $H_0$ and $H_a$. (b) decide whether to use a Wilcoxon signed-rank test or a Wilcoxon rank sum test. (c) find the critical value(s). (d) find the test statistic. (e) decide whether to reject or fail to reject the null hypothesis. (f) interpret the decision in the context of the original claim. A physician wants to determine whether an experimental medication affects an individual’s heart rate. The physician randomly selects 15 patients and measures the heart rate of each. The subjects then take the medication and have their heart rates measured after one hour. The table shows the results. At $\alpha=0.05$, can the physician conclude that the experimental medication affects an individual’s heart rate?

$$\begin{matrix} \text{Patient} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \text{8}\\ \text{Heart rate (before)} & \text{72} & \text{81} & \text{75} & \text{76} & \text{79} & \text{74} & \text{65} & \text{67}\\ \text{Heart rate (after)} & \text{73} & \text{80} & \text{75} & \text{79} & \text{74} & \text{76} & \text{73} & \text{67}\\ \text{Patient} & \text{9} & \text{10} & \text{11} & \text{12} & \text{13} & \text{14} & \text{15} & \text{ }\\ \text{Heart rate (before)} & \text{76} & \text{83} & \text{66} & \text{75} & \text{76} & \text{78} & \text{68} & \text{ }\\ \text{Heart rate (after)} & \text{74} & \text{77} & \text{70} & \text{77} & \text{76} & \text{75} & \text{74} & \text{ }\\ \end{matrix}$$

Use logarithmic differentiation to find the derivative of the function. $y=x^{\ln x}$

Determine the intervals where the function $f(x)=\frac{\ln x}{x}$ is increasing and where it is decreasing.