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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
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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:
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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+
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Gaussian Model or the Bell curve
Its equations would be y = ae-[(x – b)^2]/c , where a ,b and c are real numbers.
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y = ae-[(x – b)^2]/c
Let a = 4; b = 2 and c = 3. The graph will never touch the x axis.
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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
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Growth equation y = aebx
Let a = 5 and b = 2
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Decay equation y = ae-bx
Let a = 2 and b = 2
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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?
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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
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Logistic Growth Model
a, b and r are positive numbers.
a is the maximum limit of the function.
rxbe
ay
1
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Logistic Growth Model
Let a = 10, b = 4 and r = 2
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HomeworkHomework
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##18, 25, 28, 29, 18, 25, 28, 29,
35, 40 , 47, 50, 35, 40 , 47, 50,
63, 70, 74, 93 63, 70, 74, 93