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The purpose of this lab is to use Maple to study exponential, logarithmic,
and logistic functions. These are used to model many types of growth, as
well as in many scales, such as the Richter and decibel scales.
Logarithmic Scale: The decibel scale is a logarithmic scale used to measure sound. Measured at some point A, the measurement of the sound in decibels is
where I(A) is the sound intensity at the point A, and I0 is a reference intensity chosen so that the threshold of human hearing is approximately 0 decibels.
Exponential growth: The simplest model for growth is exponential, where it is assumed that y'(t) is proportional to y. That is,
Separating the variables and integrating (see section 4.4 of the text), we have
(If y is a positive quantity, we may drop the absolute value signs around y.) Solving for y
y = ekt + Cwhich we may write in the form y = Aekt, where A is an arbitrary positive constant. (Why?)
Exponential Decay: In a sample of a radioactive material, the rate at which atoms decay is proportional to the amount of material present. That is,
where k>0 is a constant. This is the same equation as in exponential growth, except that -k replaces k. The solution is
A(t) = A0 e-ktwhere A0 is a positive constant. Physically, A0 is the amount of material present at t=0.
Radioactivity is often expressed in terms of an element's half-life.
For example, the half-life of carbon-14 is 5730 years. This statement means
that for any given sample of , after 5730 years, half of it
will have undergone decay.
So, if the half-life is of an element Z is c years, it must be
that , so that and .
Logistic growth: One of the consequences of exponential growth is that as . However, in some situations there is a limit B to how large y can get. For example, the population of bacteria in a laboratory culture, where the food supply is limited. In such situations, the rate of growth slows as the population reaches the carrying capacity. One useful model is the logistic growth model. It assumes that the rate of growth is proportional to the product of the population and the difference between the population and its upper limit. Thus we model the growth with the differential equation
In the exercises you will use Maple to solve this equation and work with an example.
where M is the magnitude of the earthquake, x is the amplitude of the largest seismic wave as measured on a standard seismograph 100 kilometers from the epicenter and c is the amplitude of a reference earthquake of amplitude 1 micron ( 1 micron is 0.001 mm) on a standard seismograph at the same distance from the epicenter.
Separating the variables yields the equation
where A is some positive constant.