Section 3.5Section 3.5

Exponential and Logarithmic Exponential and Logarithmic ModelsModels

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

Section 3.5 Mathematical Section 3.5 Mathematical ModelsModels

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.

HomeworkHomework

P. 232 – 236 P. 232 – 236

28, 29, 32, 37, 39, 4128, 29, 32, 37, 39, 41