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Section 3.5

Section 3.5. Exponential and Logarithmic Models. Important Vocabulary. Bell-shaped curve The graph of a Gaussian model. Logistic curve A model for describing populations initially having rapid growth followed by a declining rate of growth.

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Section 3.5

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  1. Section 3.5 Exponential and Logarithmic Models

  2. Important Vocabulary • Bell-shaped curve The graph of a Gaussian model. • Logistic curve A model for describing populations initially having rapid growth followed by a declining rate of growth. • Sigmoidal curve Another name for a logistic growth curve.

  3. The exponential growth model The exponential decay model The Gaussian model The logistic growth model Logarithmic models Y = aebx Y = ae-bx y = ae-(x-b)^2/c Y = a/(1 + be-rx) Y = a + b ln x and y = a + b log10 x 5 Most Common Types of Models

  4. Section 3.5 Mathematical Models

  5. Exponential Growth • Suppose a population is growing according to the • model P = 800e0.05t, where t is given in years. • (a) What is the initial size of the population? • (b) How long will it take this population to double?

  6. Carbon Dating Model • To estimate the age of dead organic matter, scientists use the carbon dating model R = 1/1012e−t/8245 , which denotes the ratio R of carbon 14 to carbon 12 present at any time t (in years).

  7. Exponential Decay • The ratio of carbon 14 to carbon 12 in a fossil is R = 10−16. Find the age of the fossil.

  8. Gaussian Models • The Gaussian model is commonly used in probability and statistics to represent populations that are normally distributed . • On a bell-shaped curve, the average value for a population is where the maximum y-value of the function occurs.

  9. Gaussian Models • The test scores at Nick and Tony’s Institute of Technology can be modeled by the following normal distribution y = 0.0798e-(x-82)^2/50 where x represents the test scores. • Sketch the graph and estimate the average test score.

  10. Logistic Growth Models • Give an example of a real-life situation that is modeled by a logistic growth model. • Use the graph to help with the explanation.

  11. Logistic Growth Examples • The state game commission released 100 pheasant into a game preserve. The agency believes that the carrying capacity of the preserve is 1200 pheasants and that the growth of the flock can be modeled by p(t) = 1200/(1 + 8e-0.1588t) where t is measured in months. How long will it take the flock to reach one half of the preserve’s carrying capacity?

  12. Logarithmic Models • The number of kitchen widgets y (in millions) demanded each year is given by the model y = 2 + 3 ln(x + 1) , where x = 0 represents the year 2000 and x ≥ 0. Find the year in which the number of kitchen widgets demanded will be 8.6 million.

  13. Homework • P. 232 – 236 28, 29, 32, 37, 39, 41

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