Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patient's body decreases at a rate proportional to the quantity remaining. The half-life of warfarin in the body is 37 hours. (a) Sketch the quantity, Q, of warfarin in a patient's body as a function of the time, t, since stopping administration of the drug. Mark the 37 hours on your graph. (b) Write a differential equation satisfied by Q. (c) How many days does it take for the drug level in the body to be reduced to 25% of the original level?

The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5730 years. Let C(t) be the amount of carbon-14 present at time t. (a) Find the value of the constant k in the differential equation C' = -kC. (b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material. How old was the Shroud of Turin at the time of this data?

Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patient's body decreases at a rate proportional to the quantity remaining. The half-life of warfarin in the body is 37 hours. Sketch the quantity, Q, of warfarin in a patient's body as a function of the time, t, since stopping administration of the drug. Mark the 37 hours on your graph .

The number of fish, P, in a lake with an initial population of 10,000 satisfies, for constant k and r, $\frac{d P}{d t}=k P-r.$ If k = 0.05, what must r be in order for the fish population to remain at a constant level?

Money in a bank account grows continuously at an annual rate of r (when the interest rate is 5%, r=0.05, and so on). Suppose $2000 is put into the account in 2010. Solve the differential equation.

Classify each of the following substances as a nonelectrolyte, weak electrolyte, or strong electrolyte in water:

$$\mathrm { H } _ { 2 } \mathrm { SO } _ { 3 }$$

(a) Find the equilibrium solution to the differential equation $\frac{d y}{d t}=0.5 y-250.$ (b) Find the general solution to this differential equation. (c) Sketch the graphs of several solutions to this differential equation, using different initial values for y. (d) Is the equilibrium solution stable or unstable?

At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is 68$^{\circ} \mathrm{F}$ in your house. At 10:00 pm, it is 57$^{\circ} \mathrm{F}$ in the house, and you notice that it is 10$^{\circ} \mathrm{F}$ outside. Assuming that the temperature, T, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by T.

At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $68^{\circ} \mathrm{F}$ in your house. At 10:00 pm, it is $57^{\circ} \mathrm{F}$ in the house, and you notice that it is $10^{\circ} \mathrm{F}$ outside. Assuming that the temperature, $T$, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by $T$.

Prove or disprove. For every positive rational number b, there exists an irrational number a with 0 < a < b.

Currently, the Smiths have $50,000 to invest for 18 months. They have two options open to them: (a) Invest the money in a certificate paying interest at the nominal rate of 5% compounded quarterly; (b) Invest the money in a savings account earning interest at the annual rate of 4.5% compounded continuously. How much money will they have in 18 months with each option?

At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $68^{\circ} \mathrm{F}$ in your house. At 10:00 pm, it is $57^{\circ} \mathrm{F}$ in the house, and you notice that it is $10^{\circ} \mathrm{F}$ outside. Assuming that the temperature, $T$, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by $T$.

What assumption did you make in above Problem about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? Why?

A cup of coffee is made with boiling water and stands in a room where the temperature is $20^{\circ} \mathrm{C}$. If $H(t)$ is the temperature of the coffee at time $t$, in minutes, explain what the differential equation

$$\frac{d H}{d t}=-k(H-20)$$

says in everyday terms. What is the sign of $k$ ?

The following sentences contain errors in capitalization. Circle each incorrectly used lowercase letter.

The national aeronautics and space administration was formed in 1958.