 CALCULUS Understanding Its Concepts and Methods

Exponential growth and decay, Doubling time, Half life, Logistic model, Newton's law of cooling

The differential equation Dt u = ku has the variables u and t on the left, and the variable u on the right. The general solution is u(t) = Cekt. A positive value of k yields exponential growth and a negative value of k yields exponential decay. The rates of growth and decay are described by the doubling time and half life, respectively.

As a first approximation, the rate of change of a population  P is proportional to P itself.

The logistic model is given by the equation

Dt P = k(1 - P/M)

where M is the largest population that the environment will sustain.

Newton's law of cooling says that the rate of change in the temperature of a small object is proportional to the difference in temperature between the object and its surroundings. The temperature of a cup of coffee drops quickly when the coffee is hot. As it cools, the rate of cooling slows.

A differential equation that describes this law is

Dt C = k(C - T)

where C is the temperature of the object and  T is the surrounding temperature.

Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.