3.5 Exponential and Logarithmic Models Gaussian Model Logistic Growth model Exponential Growth and Decay

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:continuous probability distributionrandom variablesmeanprobability density functionGaussian function

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 equationy = ae bx b> 0 Decay equationy = ae -bx b>0 Both these models we have seen before in Algebra 2 and in Pre- Cal

Growth equationy = ae bx Let a = 5 and b = 2

Decay equationy = 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. sigmoid curvePierre François Verhulstexponential Pierre Francois Verhuist

Logistic Growth Model a, b and r are positive numbers. a is the maximum limit of the function.

Logistic Growth Model Let a = 10, b = 4 and r = 2

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