3.5 Exponential and Logarithmic Models

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3.5 Exponential and Logarithmic Models

• Gaussian Model
• Logistic Growth model
• Exponential Growth and Decay

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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 bgt 0
• Decay equation y ae-bx bgt0
• 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_functi
  on

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Logistic Growth Model

• a, b and r are positive numbers.
• a is the maximum limit of the function.

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Logistic Growth Model

• Let a 10, b 4 and r 2

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Homework

• Page 243- 248
• 18, 25, 28, 29,
• 35, 40 , 47, 50,
• 63, 70, 74, 93