A certain piece of news is being broadcast to a potential audience of 200,000 people. Let f(t) be the number of people who have heard the news after t hours. Suppose that y=f(t) satisfies

$$y^{\prime}=.07(200,000-y), \quad y(0)=10 .$$

Describe this initial-value problem in words.

Consider a certain commodity that is produced by many companies and purchased by many other firms. Over a relatively short period there tends to be an equilibrium price $p_0$ per unit of the commodity that balances the supply and the demand. Suppose that, for some reason, the price is different from the equilibrium price. The Evans price adjustment model says that the rate of change of price with respect to time is proportional to the difference between the actual market price p and the equilibrium price. Write a differential equation that expresses this relation. Sketch two or more solution curves.

The differential equation $y^{\prime}=2 t y+e^{t^2}, y(0)=5$, has solution $y=(t+5) e^{t^2}$. In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calculated from the solution. What is the greatest difference between corresponding values in the second and third rows?

$t_i$	0	.2	.4	.6	.8	1	1.2	1.4	1.6	1.8	2
$y_i$	5
y	5	5.412									382.2

In Exercise given below, solve the given equation using an integrating factor. Take t>0.

$$y^{\prime}=e^{-t}(y+1)$$

Consider an investment that pays off $800 or$1,400 per $1,000 invested with equal probability. Suppose you have$1,000 but are willing to borrow to increase your expected return. What would happen to the expected value and standard deviation of the investment if you borrowed an additional $1,000 and invested a total of$2,000? What if you borrowed $2,000 to invest a total of$3,000?

Derive the formula for the population in Example mentioned, if the population in 1995 was 2 million. (The formula is given following the solution of the said Example.)

Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the coin comes up heads with probability 2/3.

b. Compute the expected value of the game.

Solve the following differential equations:

$$y^2 y^{\prime}=\tan t$$

Let f(t) be the balance in a savings account at the end of t years. Suppose that y=f(t) satisfies the differential equation $y^{\prime}=.04 y+2000$. (a) If after 1 year the balance is $10,000, is it increasing or decreasing at that time? At what rate is it increasing or decreasing at that time? (b) Write the differential equation in the form$y^${\prime}$=k(y+M)$. (c) Describe this differential equation in words.

One or more initial conditions are given for the differential equation in the this exercise. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant solutions on the ty-graph whether they are mentioned or not.

The graph of $z=-\frac{1}{2} y \ln (y / 30)$ has relative maximum point at $y \approx 11.0364$ and y-intercept at y-30; y(0)-1, y(0)=20, and y(0)=40. Sketch the solutions to the Gompertz growth equation

$$\frac{d y}{d t}=-\frac{1}{2} y \ln \frac{y}{30} .$$

Money left in a savings account grows exponentially with time. Suppose that you invest $1000 and find that a year later you have$1100 in your account.

In how many years will your investment double?

Which of the following scenarios best illustrates how effective or ineffective maintenance rehearsal is in transferring information into LTM?

Review in the Grammar/Mechanics Handbook. Then select the correctly punctuated sentence and record its letter in the space provided. Also record the appropriate G/M guideline to illustrate the principle involved. When you finish, compare your responses with those provided at the bottom of the page. If your answers differ, study carefully the appropriate guideline.

a. Apple earns most of its income from the following: Macs, iPads, and iPhones.

b. Apple earns most of its income from the following; Macs, iPads, and iPhones.

Assume that all numbers are approximate unless stated otherwise.
Certain types of iPhones and iPads weigh approximately $129$ g and $298.8$ g, respectively. What is the total weight of $12$ iPhones and $16$ iPads of these types?