The number of Medicare-certified home-health-care agencies (70% are freestanding, certified home-health-care agencies (70% are freestanding, been declining at the rate of $0.186 e^{-0.02 t}$ $0.186 e^{-0.02 t}$ thousand agencies/year between 1988 (t = 0) and 2002 (t = 14). The number of such agencies stood at 9.3 thousand in 1988. Find an expression giving the number of health-care agencies in year t.

Find the steady-state vector.

$$\left[\begin{array}{ll} .1 & .5 \\ .9 & .5 \end{array}\right]$$

The U.S. population is approximated by the function $P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}}$ where P(t) is measured in millions of people and t is measured in 30-year intervals, with t = 0 corresponding to 1930. What is the expected population of the United States in 2020 (t=3)?

The growth rate of the bacterium Escherichia coli, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 20 min. a. If the initial cell population is 100, determine the function Q(t) that expresses the exponential growth of the number of cells of this bacterium as a function of time t (in minutes). b. How long will it take for a colony of 100 cells to increase to a population of 1 million? If the initial cell population were 1000, how would this alter our model?

Find the derivative of the function. $f(x)=3 x^{2} \ln 2 x$

The methods of this section apply to physical systems with more than one degree of freedom. Suppose there are two displacements, $y_{1}$ and $y_{2}$, with associated velocities $v_{1}=y_{1}^{\prime}$ and $v_{2}=y_{2}^{\prime}$, and suppose that the system is modeled by the system of equations $\mathbf{y}^{\prime \prime}=\mathbf{f}(\mathbf{y})$, where $\mathbf{y}=\left(y_{1}, y_{2}\right)^{T}$, and the forcing term is $\mathbf{f}=\left(f_{1}, f_{2}\right)^{T}$. We also put the velocities into the vector $\mathbf{v}=\left(v_{1}, v_{2}\right)^{T}$. Then we can rewrite the second order system as the first-order system

$$\begin{array}{l} \mathbf{y}^{\prime}=\mathbf{v} \\ \mathbf{v}^{\prime}=\mathbf{f}(\mathbf{y}). \end{array}$$

We will say that this system is conservative if there is a function $U(\mathrm{y})$ such that $\partial U / \partial y_{1}=-f_{1}$ and $\partial U / \partial y_{2}=-f_{2}$. Show that if the system is conservative, then the total energy $E(y, v)=\frac{1}{2}|\mathbf{v}|^{2}+U(\mathbf{y})=\frac{1}{2}\left(v_{1}^{2}+v_{2}^{2}\right)+U(\mathbf{y})$ is a conserved quantity for the first-order system.

For the following exercise, for each linear equation, a. give the slope m and y-intercept b, if any, and b. graph the line. 4y + 24 = 0

Prove that if p is a prime and P is a non-abelian group of order $p^{3}$ then $\Phi(P)=Z(P)$ and

$$P / \Phi(P) \cong Z_{p} \times Z_{p}.$$

Deduce that P has p + 1 maximal subgroups.

For the following exercise, for each pair of functions, find a. f + g b. f - g c. $f \cdot g$ d. f/g. Determine the domain of each of these new functions. $f(x)=x-8, g(x)=5 x^{2}$

Prove that if D is a division ring then C(a) is a division ring for all $a \in D$.

A woman getting dressed up for a night out is asked by her significant other to wear a red dress, high-heeled sneakers, and a wig. In how many orders can she put on these objects?

The Major League Baseball team with the lowest payroll in 2014 was the Houston Astros. The following table gives the salary of each Astro in 2014.

$$\begin{array}{lc} \text { Name } & \text { Salary } \\ \hline \text { Scott Feldman } & \$ 12,000,000 \\ \text { Jose Veras } & \$ 3,850,000 \\ \text { Chad Qualls } & \$ 2,750,000 \\ \text { Jason Castro } & \$ 2,450,000 \\ \text { Jose Altuve } & \$ 1,437,500 \\ \text { Jesus Guzman } & \$ 1,300,000 \\ \text { Matt Dominguez } & \$ 510,100 \\ \text { Chris Carter } & \$ 510,000 \\ \text { Dallas Keuchel } & \$ 508,700 \\ \text { Josh Fields } & \$ 506,500 \\ \text { Carlos Corporan } & \$ 505,300 \\ \text { Anthony Bass } & \$ 505,200 \\ \text { Marwin Gonzalez } & \$ 504,500 \\ \text { Robbie Grossman } & \$ 504,500 \\ \text { Brad Peacock } & \$ 504,300 \\ \text { L. J. Hoes } & \$ 502,900 \\ \text { Jared cosart } & \$ 500,000 \end{array}$$

(a) Find the mean, median, and mode of the salaries. (b) Which average best describes these data? (c) Why is there such a difference between the mean and median?

(a) find the intervals where the function f is increasing and where it is decreasing, (b) find the relative extrema of f, (c) find the intervals where the graph of f is concave upward and where it is concave downward, and (d) find the inflection points, if any, of f. $f(x)=(x-2)^{3}$

Verify that the Fundamental theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. $\int_{C} \nabla\left(1+x^{2} y z\right) \cdot d \mathbf{r}$, where C is the helix r(t) = (cos 2t, sin 2t, t), for $0 \leq t \leq 4 \pi$