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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

Section 3.5

Exponential and Logarithmic Models

important vocabulary
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.
5 most common types of models
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
exponential growth
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?
carbon dating model
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).
exponential decay
Exponential Decay
  • The ratio of carbon 14 to carbon 12 in a fossil is R = 10−16. Find the age of the fossil.
gaussian models
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.
gaussian models1
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.
logistic growth models
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.
logistic growth examples
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?

logarithmic models
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.

homework
Homework
  • P. 232 – 236

28, 29, 32, 37, 39, 41