3.5 Exponential and Logarithmic Models

1. 3.5 Exponential and Logarithmic Models Gaussian Model Logistic Growth model Exponential Growth and Decay

2. 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:

3. 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+

4. Gaussian Model or the Bell curve Its equations would be y = ae-[(x –b)^2]/c , where a ,b and c are real numbers.

5. y = ae-[(x –b)^2]/c Let a = 4; b = 2 and c = 3. The graph will never touch the x axis.

6. 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

7. Growth equation y = aebx Let a = 5 and b = 2

8. Decay equation y = ae-bx Let a = 2 and b = 2

9. 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?

10. 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

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

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

13. Homework Page 243- 248 # 18, 25, 28, 29, 35, 40 , 47, 50, 63, 70, 74, 93