3.5 Exponential 3.5 Exponential and Logarithmic and Logarithmic

ModelsModelsGaussian ModelGaussian Model

Logistic Growth modelLogistic Growth model

Exponential Growth and Exponential Growth and DecayDecay

Gaussian Model or the Bell curve

The normal (or Gaussian) distribution is a continuous probability distribution that is often used as a first approximation to describe real-valued random variables that tend to cluster around a single mean value. The graph of the associated probability density function is "bell"-shaped, and is known as the

Gaussian function or bell curve:

Gaussian Model or the Bell curve

If I was curving your grades, 68.2% of the students would have a C, 13.6% a B or D and 2.1% a A or F.

0.1% would have an A+

Gaussian Model or the Bell curve

Its equations would be y = ae-[(x – b)^2]/c , where a ,b and c are real numbers.

y = ae-[(x – b)^2]/c

Let a = 4; b = 2 and c = 3. The graph will never touch the x axis.

Exponential Growth/ Decay models

Growth equation y = aebx b> 0

Decay equation y = ae-bx b>0

Both these models we have seen before in Algebra 2 and in Pre- Cal

Growth equation y = aebx

Let a = 5 and b = 2

Decay equation y = ae-bx

Let a = 2 and b = 2

Will a small lake have exponential growth of game fish forever?

No,

What are the factors that keep the lake from the lake filling up with fish?

Logistic growth model• A logistic function or logistic curve is a common

sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve (abbreviated S-curve) of growth of some population P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

Pierre Francois Verhuist

http://en.wikipedia.org/wiki/Logistic_function

Logistic Growth Model

a, b and r are positive numbers.

a is the maximum limit of the function.

rxbe

ay

1

Logistic Growth Model

Let a = 10, b = 4 and r = 2

HomeworkHomework

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35, 40 , 47, 50, 35, 40 , 47, 50,

63, 70, 74, 93 63, 70, 74, 93