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?

A radioactive sample contains 1.55 g of an isotope with a half-life of 3.8 days. What mass of the isotope remains after 5.5 days?

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 $(\theta, \omega)=(0,0)$.

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.

Find the absolute maximum value and the absolute minimum value, if any, of each function. $f(x)=x^{2}-2 x-3$ on [-2, 3]

Assume that T is a linear transformation. Find the standard matrix of T. $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ first reflects points through the horizontal $x_{1}$-axis and then rotates points $-\pi / 2$ radians.

Construct the indicated confidence interval for $\mu_{1}-\mu_{2}$. Assume the populations are approximately normal with unequal variances. To compare the mean finishing times of male and female participants in a 10K race, you randomly select several finishing times from both sexes. The results are shown at the left. Construct an 80% confidence interval for the difference in mean finishing times of male and female participants in the race. Sample Statistics for Finishing Times of 10K Race Participants

$$\begin{matrix} \text{Males} & \text{Females}\\ \ {\bar{x}_{1}=0.73 \mathrm{h}} & \ {\bar{x}_{2}=0.75 \mathrm{h}}\\ \ {s_{1}=0.17 \mathrm{h}} & \ {s_{2}=0.04 \mathrm{h}}\\ \ {n_{1}=20} & \ {n_{2}=12}\\ \end{matrix}$$

Let W be the set of all vectors of the form shown, where a, b and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.

$$\left[\begin{array}{c} 1 \\ 3 a-5 b \\ 3 b+2 a \end{array}\right]$$

Markov chains have been used to study music. The relative frequency with which one note is followed by another in a particular song can be recorded in a transition matrix, which can then be used to generate random sequences of notes with the same frequency pattern. If the rhythms are also incorporated, new songs that are in some way similar to the original song can be generated. (a) Consider the simple melody "Merrily We Roll Along," with the following sequence of notes in the key of G:

$$\begin{array}{l} BAGABBB \\ AAABBB \\ BAGABBB \\ AABAG \end{array}$$

Suppose this song is played in a loop, so that the final G is followed by the initial B. Construct a transition matrix for this melody. For example, of the 12 occurrences of B, 5 are followed by A, so the entry in row B, column A, is 5/12. (b) Determine an equilibrium vector for the matrix in part (a). Then explain why, in retrospect, this result is obvious by looking at the frequency of notes in the original melody.

Let

$$G=A_{1} \times A_{2} \times \cdots \times A_{n}$$

and for each i let $B_{i}$ be a normal subgroup of $A_{i}$. Prove that

$$B_{1} \times B_{2} \times \cdots \times B_{n} \unlhd G$$

and that

$$\left(A_{1} \times A_{2} \times \cdots \times A_{n}\right) /\left(B_{1} \times B_{2} \times \cdots \times B_{n}\right) \cong\left(A_{1} / B_{1}\right) \times\left(A_{2} / B_{2}\right) \times \cdots \times\left(A_{n} / B_{n}\right)$$

Indicate whether the statement is true or false. If the percent birth rate and the percent death rate in a country are both decreasing, then the country's population is decreasing.

Identify the two events described in the study. Do the results indicate that the events are independent or dependent? Explain your reasoning. A study found that people who suffer from obstructive sleep apnea are at increased risk of having heart disease.

A micrometer is 0.000001 meter. Use your calculator to determine how many micrometers are in each of the following. Write your answer in both calculator and scientific notation. 5,000 meters