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Exponential and Logarithmic Models 3/1/2011

DO NOW:

I have a 12 inch balloon. It is doubling in size every minute.

How big will it be in 7 minutes?

How many minutes until it is a mile long? Psst. A mile is 5,280 feet

SWBAT:Use exponential growth and decay functions to solve real-life problemsUse logistic growth functions to solve real-life problemsUse logarithmic functions to solve real-life problems

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Common Mathematical Models

Many business applications and natural phenomena

can be modeled by exponential and logarithmic

functions. In this section we will explore logistic

models and maybe even solve some real-life

applications

SWBAT:Use exponential growth and decay functions to solve real-life problemsUse logistic growth functions to solve real-life problemsUse logarithmic functions to solve real-life problems

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Applications of Logistic Growth Models

Logistic growth functions are used to model situations where

initially there is an increasing rate of growth followed by a

decreasing rate of growth.

Some common examples include the spread of a disease

within a population and the growth of certain populations. In

these cases, the functions have an upper bound which is equal

to the maximum population capacity. This upper bound and the

line

y = 0 are the horizontal asymptotes of the graphs of logistic

functions.

SWBAT: Use exponential growth and decay functions to solve real-life problems

Use logistic growth functions to solve real-life problems

Use logarithmic functions to solve real-life problems

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Graph of Logistic Model

Domain: (-, ) Range: (0, c) Horizontal Asymptotes: y = 0, y = c

SWBAT: Use exponential growth and decay functions to solve real-life problems Use logistic growth functions to solve real-life problems Use logarithmic functions to solve real-life problems

Horizontal asymptote y=c

provides a limit to growth.

1 bx

cy

ae

Original amount at x=0

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A Logistic Growth Model

Example: Fruit flies are placed in a half-pint milk bottle with a banana

(for food) and yeast plants (for food and to provide stimulus to lay

eggs). Suppose that the fruit fly population P after t days is given

by

a) What is the carrying capacity of the half-pint bottle? (That is, what is the upper limit of the population?)

b) How many fruit flies were initially placed in the half-pint bottle?

c) When will the population be 180?

SWBAT: Use exponential growth and decay functions to solve real-life problems

Use logistic growth functions to solve real-life problems

Use logarithmic functions to solve real-life problems

0.37

230( )

1 56.5 tP t

e

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A Logistic Growth Model (continued)

Solution:

a) We can use the TABLE feature on the graphing calculator

table to find P(t) as t→.

P approaches 230 as t gets increasingly larger.

Notice, the LIMITING CAPACITY IS THE NUMERATOR OF THIS FUNCTION. It is also the upper horizontal asymptote of the graph of this function. (y = 0 is the lower asymptote).

So, the carrying capacity of the bottle is _______ fruit flies.

y = 230

y = 0

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A Logistic Growth Model (continued)

Solution:

b) To find the initial number of fruit flies in the bottle, we need

to find P(0).

Algebraically: Graphically:

Initially, there were ________ fruit flies in the half-pint bottle.

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A Logistic Growth Model (continued)

Solution:

c) To find when the population will reach 180, set P(t)=180.

Algebraically: Graphically:

It will take approx. _______ days for the pop. to reach 180 fruit flies.

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Applications of Logarithmic Models

Many relations between variables are best modeled by a logarithmic

function.

Some common examples include:

the relation between an earthquake’s magnitude and intensity on

the Richter scale,

the relation between atmospheric pressure and height,

the relation between sound level (in decibels) and intensity,

as well as many economic models.

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POINT OF INFLECTION: is a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along a winding road, inflection is the point at which the steering-wheel is momentarily "straight" when being turned from left to right or vice versa. With Logistic models, it occurs

where the graph ‘levels off’; it’s where the

rate of increase is at it’s peak before it

‘slows down’ The End