The number of US. citizens aged 65 years and older from 1900 through 2050 is estimated to be growing at the rate of $R(t)=0.063 t^{2}-0.48 t+3.87$ $(0 \leq t \leq 15)$ million people/decade, where t is measured in decades, with t=0 corresponding to 1900. Show that the average rate of growth of U.S. citizens aged 65 years and older between 2000 and 2050 will be about twice the average rate between 1950 and 2000.

Prove that any finite group has a finite exponent. Give an example of an infinite group with finite exponent. Does a finite group of exponent m always contain an element of order m?

The millions of U.S. citizens age 65 and older with Alzheimer's disease from 2000 and projected to 2050 . If the number of millions of these U.S. citizens is the function f(x), where x is the year, use the graph to answer the following. In what year do we expect 11 million U.S. citizens age 65 and older to have Alzheimer’s disease?

Cell phone advertising spending is expected to grow at the rate of $R(t)=0.8256 t^{-0.04} \ (1 \leq t \leq 5)$ billion dollars/year between 2007 (t = 1) and 2011 (t = 5). The cell phone ad spending in 2007 was $0.9 billion. a. Find an expression giving the cell phone ad spending in year t. b. If the trend continued, what was the cell phone ad spending in 2014?

Cell-phone ad spending between 2005 (t=1) and 2011 (t=7) is projected to be $S(t)=0.86 t^{0.96}$ $(1 \leq t \leq 7)$ where S(t) is measured in billions of dollars and t is measured in years. What is the projected average spending per year on cell-phone ads between 2005 and 2011?

Carl Williams, the proprietor of Carl Williams Auto Sales, estimates that with extensive television advertising, car sales over the next several years could be increasing at the rate of $5 e^{0.3 t}$ thousand cars/year, 1 years from now, instead of at the current rate of $5+0.5 t^{3 / 2}$ thousand cars/year, t years from now. Find how many more cars Carl expects to'sell over the next 5 years by implementing his advertising plans.

The management of Ditton Industries has determined that the daily marginal revenue function associated with selling x units of their deluxe toaster ovens is given by $R^{\prime}(x)=-0.1 x+40$ where $R^{\prime}(x)$ is measured in dollars per unit. Find the daily total revenue realized from the sale of 200 units of the toaster oven.

Model each application with a differential equation. All rates are assumed to be with respect to time. The voltage drop across an inductor is proportional to the rate at which the current is changing with respect to time.

US. Internet advertising revenue grew at the rate of $R(t)=0.82 t+1.14$ $(0 \leq t \leq 4)$ billion dollars/year between 2002 (t=0) and 2006 (t=4). The advertising revenue in 2002 was $5.9 billion. Find an expression giving the advertising revenue in year t.

US. Internet advertising revenue grew at the rate of $R(t)=0.82 t+1.14$ $(0 \leq t \leq 4)$ billion dollars/year between 2002 (t=0) and 2006 (t=4). The advertising revenue in 2002 was $5.9 billion. If the trend continued, what was the Internet advertising revenue in 2007?

To test an individual’s use of calcium, a researcher injects a small amount of radioactive calcium into the person’s bloodstream. The calcium remaining in the bloodstream is measured each day for several days. Suppose the amount of the calcium remaining in the bloodstream (in milligrams per cubic centimeter) $t$ days after the initial injection is approximated by $C(t)=\frac{1}{2}(2 t+1)^{-1 / 2}$. Find the rate of change of the calcium level with respect to time for the following numbers of days.

$$\begin{array}{lll}{\text { a.0 }} & {\text { b.4 }} & {\text { c. } 7.5}\end{array}$$

d. Is $C$ always increasing or always decreasing? How can you tell?

The U.S. national debt stood at $10.025 trillion in 2008 (t = 0) and was growing at the rate of$R(t)=0.070251 t^{2}-0.51548 t+2.1667 \ (0 \leq t \leq 4)$ trillion dollars/year in year t between 2008 and 2012 What was the U.S. national debt in 2012?

Diagonalize the matrices if possible. The real eigenvalues are included below the matrix.

$$\begin{array}{c} {\left[\begin{array}{cc} 2 & 0 & -2 \\ 1 & 3 & 2 \\ 0 & 0 & 3 \end{array}\right]} \\ \lambda=2,3 \end{array}$$

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.