Section 3.5 Exponential and Logarithmic Models. Important Vocabulary Bell-shaped curve The graph of...
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Transcript of Section 3.5 Exponential and Logarithmic Models. Important Vocabulary Bell-shaped curve The graph of...
Important VocabularyImportant 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 Models5 Most Common Types of Models
The exponential growth model
The exponential decay model
The Gaussian model
The logistic growth model
Logarithmic models
Y = aeY = aebxbx
Y = aeY = ae-bx-bx
y = aey = ae-(x-b)^2/c-(x-b)^2/c
Y = a/(1 + beY = a/(1 + be-rx-rx))
Y = a + b ln x andY = a + b ln x and y = a + b logy = a + b log1010 x x
Exponential GrowthExponential 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 ModelCarbon Dating Model
To estimate the age of dead organic matter, scientists use the carbon dating model R = 1/1012 e−t/8245 , which denotes the ratio R of carbon 14 to carbon 12 present at any time t (in years).
Exponential DecayExponential Decay
The ratio of carbon 14 to carbon 12 in a fossil is R = 10−16. Find the age of the fossil.
Gaussian ModelsGaussian 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 ModelsGaussian Models
The test scores at Nick The test scores at Nick and Tony’s Institute of and Tony’s Institute of Technology can be Technology can be modeled by the modeled by the following normal following normal distribution y = distribution y = 0.0798e0.0798e-(x-82)^2/50-(x-82)^2/50 where where x represents the test x represents the test scores. scores.
Sketch the graph and Sketch the graph and estimate the average estimate the average test score.test score.
Logistic Growth ModelsLogistic 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 ExamplesLogistic Growth Examples The state game commission The state game commission
released 100 pheasant into a released 100 pheasant into a game preserve. The agency game preserve. The agency believes that the carrying believes that the carrying capacity of the preserve is capacity of the preserve is 1200 pheasants and that the 1200 pheasants and that the growth of the flock can be growth of the flock can be modeled by modeled by
p(t) = 1200/(1 + 8ep(t) = 1200/(1 + 8e-0.1588t-0.1588t) where ) where t is measured in months. t is measured in months.
How long will it take the flock to How long will it take the flock to reach one half of the preserve’s reach one half of the preserve’s carrying capacity?carrying capacity?
Logarithmic ModelsLogarithmic 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.