Week+26+March+17-20

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Tuesday March 17
For most of the period we went over the 2002 Free Response problems we had turned in before Spring Break. Our assignment was Exploration set 7.1, an introduction to exponential growth and decay and differential equations.The exploration will appear shortly.

Wed. March 18
First order, differential equations are equations describing the rate of change of a function in terms of the function itself. If the equation is separable, it is possible to separate common differentials and variables on the same side of the equals sign. Using two examples, the process for writing and solving these differential equations is shown below.



Separate the variables so that the terms in P are with the dP and all other terms are with the dt

Now integrate both sides.

We only need one constant of integration.



Thurs March 19.
Thus far, all our solutions to the differential equations have been of the form P(t) = Ae^(kt) which is not the whole story for differential equations. Today we solved differential equations which had solutions of varied form.To introduce the idea, we used exploration 7.3a which led us through a memory retention problem.

Ira Member is a freshman at a large university. One evening he attends a reception at which there are many members of his class whom he has not met. He wants to predict how many new names he will remember at the end of the reception.

1. Ira assumes that he meets people at a constant rate of R people per hour. Unfortunately, he forgets names at a rate proportional to y, the number he remembers. The more he remembers, the faster he forgets! Let t be the number of hours he has been at the reception. What does dy/dt equal? (Use the letter k for the proportionality constant. 2. The equation in problem 1 is a **differential equation** because it has differentials in it. By algebra, separate the variables so that all terms containing y appear on one side of the equation and all terms containing t appear on the other side. 3. Integrate both sides of the equation in problem 2. 4. Show that the solution in problem 3can be transformed into the form ky = R - Ce^(-kt) where C is a constant related to the constant of integration. Explain what happens to the absolute value sign that you got from integrating the reciprocal function. 5. Use the initial condition y = 0 when t = 0 to evaluate the constant C. 6. Suppose that Ira meets 100 people per hour, and that he forgets at a rate of 4 names per hour when y = 10 names. Write the particular equation expressing y in terms of t. 7. How many names will Ira have remembered at the end of the reception when t = 3 hours?

Fri March 20th
Today we were given four differential equation problems to solve.

1. Suppose a deer population grows exponentially at a rate of 10% per year. To keep the population size reasonable, the Park Service removes 20 deer each year. a) Write a differential equation for the rate of change of the size of the deer population with respect to time. b) Solve the differential equation.

2. A chain smoker smokes 5 cigarettes every hour.From each cigarette, 0.4 mg of nicotine is absorbed into the person's bloodstream. Nicotine leaves the body at a rate proportional to the amount present, with constant of proportionality -0.346. a) Write a differential equation for the level of nicotine in the body, N, in mg, as a function of time, t, in hours.

b) Solve the differential equation. Assume that initially there is no nicotine in the blood.

c) The person wakes up at 7 a.m. and begins smoking. How much nicotine is in the blood when the person goes to sleep at 11 p.m.? N(16) = 5.758 so the person has approximately 5.8 mg of nicotine in the blood by 11 pm

3. Dead leaves accumulate on the ground in a forest at a rate of 3 grams per square centimeter per year. At the same time, theses leaves decompose at a continuous rate of 75% per year. Write a differential equation for the total quantity of dead leaves (per square centimeter) at time t. Sketch a solution showing that the quantity of dead leaves tends towards an equilibrium level. What is the equilibrium level?

4 According to a simple physiological model, an athletic adult needs 20 calories per day per pound of body weight to maintain his weight. If he consumes more or fewer than those required to maintain his weight,his weight will change at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight; the constant of proportionality is 1/3500 pounds per calorie. Suppose that a particular person has a caloric intake of I calories per day. Let W(t) be the person's weight in pounds at time t (measured in days). a) What differential equation has solution W(t)? b) Solve this differential equation. c) Draw a graph of W(t) if the person starts out weighing 160 pounds and consumes 3000 calories a day. Label the axes and any intercepts and asymptotes clearly.