Lecture Notes: Monday, November 2, 1998

Eighth Homework Assignment - for discussion at recitations on Tuesday November 3, and Thursday November 5.

Chapter 5

Section 6 (Exponential Models)

Exponential Growth. Rate of change is proportional to amount present.

(dy/dt) = ky.

Claim y = A*exp(kt) for some constant A.

Proof:

Let y satisfy the equation. Consider g = y/exp(kt). We showed g' = 0, so g is constant. Done.

Examples:

• 5.6 #4 World Population growth.
• Old Final Exam Spring 1997 #7.
• 5.6 #10 Nuclear Fallout
• 5.6 #11 Carbon-14 dating.

Other Exponential Examples:

Newton's Law of Cooling, and diffusion of information by mass media.

In both cases, the rate at which a certain quantity is changing is proportional to the difference between that quantity and some limiting level for that quantity.

(dy/dt) = k(T - y)

Let y satisfy the equation. Let g = (y - T)/exp(-kt). We showed g' = 0, so g = constant. From this we concluded that

y = T + A*exp(-kt), for some constant A.

As a check, note that y approaches the limiting level T as time goes to plus infinity.

We will work some examples of this sort on Friday.

MAPLE Demonstration #9 Please work through this demonstration. It uses MAPLE'S differential equation solver on some problems of this sort.

More Exponential Models:

Epidemics, Spread of rumors, growth of populations with environmental limits.

In these situations a quantity is changing at a rate proportional to the amount at the given time, and proportional to the difference between the amount and a limiting level. The typical equation is:

(dy/dt) = (k/P)*y*(P - y).

The solution in this case is called the Logistic Curve. We do not have the tools to derive it. The result is:

y = P/(1 + B*exp(-kt))

Example: Flu hits a city of 1,000,000 people. At the beginning of monitoring, 200 cases are known. After the first week, there are 500 more cases. How many will have become ill with the flu by the end of 10 weeks? Ans - 982,242. We looked at this result in MAPLE.

Other examples:

• 5.6 #20 Spread of a Rumor.
• 5.6 # 18 Growth of a population with environmental limits. (We did not have time to do this one. Please try it on your own.)

MAPLE Demonstration #10 Please work through this demonstration. It uses MAPLE'S differential equation solver on some problems of this sort.

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