Section 3.5

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# Section 3.5 - PowerPoint PPT Presentation

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

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.

• Sigmoidal curve Another name for a logistic growth curve.
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
• 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
• 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
• The ratio of carbon 14 to carbon 12 in a fossil is R = 10−16. Find the age of the fossil.
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 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
• 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
• 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
• 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
• P. 232 – 236

28, 29, 32, 37, 39, 41