Exponential Growth. Definition: Growing without bound. Ie. Nothing limits the growth.
-
Upload
sophie-reeves -
Category
Documents
-
view
229 -
download
0
Transcript of Exponential Growth. Definition: Growing without bound. Ie. Nothing limits the growth.
Exponential Growth
Definition:
Growing without bound.
Ie. Nothing limits the growth.
What does it look like?
5
4
3
2
1
-1
-2
-4 -2 2 4
Population
Independent Axis
Dependent Axis
Population is a function of time:
P t( ) =P0ekt
Where:
P t( ) is the population after t years;P0 is the initial population;k is the growth rate constant;t is the time in years.
Population:
Write down the first thing you think of when you hear the word “population”.
Not a definition, just the first thing that pops into your head.
Population:
A population can be:
People
Rust
Bacteria
Rocks
Anything that can be grouped and categorized.
Example
A population of people increases from 2500 to 2700 in 25 years. What is the growth rate constant for this population?
Example
A population of bacteria doubles every 43 minutes. What is the growth rate constant?
Logistic Growthand
Exponential Decay
Logistic Growth
Logistic growth is where a population is growing toward a maximum number that is allowed for that population.
In other words, a population that is growing logistically has limitations.
What does it look like?
60
50
40
30
20
10
-10
-40 -20 20 40 60
Populations
Populations can’t grow exponentially forever.
What factors inhibit growth?
Factors that inhibit growth
Size of the area (overpopulation)
Disease
Predators
Natural Disasters
Man-made disasters
War
Limit of Growth
When a population reaches its maximum number, we say that the population has met its:
Carrying capacity
Logistic Modeling
The growth model for a population that grows logistically is:
P t( ) =N
1−ce−kt
Example
The number of fruit flies in a population after t days can be found by the equation:
P t( ) =230
1+ 56.5e−0.37t
Example (cont.)
What is the carrying capacity?
500
450
400
350
300
250
200
150
100
50
-50
5 10 15 20 25
f x( ) = 230
1+56.5 ⋅e-0.37 ⋅x
Example (cont.)
What is the initial population?
P t( ) =230
1+ 56.5e−0.37t
Example (cont.)
What will the population be after 15 days? After 30 days?
P t( ) =230
1+ 56.5e−0.37t
Decay
Exponential decay is normally discussed in terms of radioactive decay.We model decay with the equation:
P t( ) =P0ekt
Decay
Decay is normally discussed in reference to a half-life.
A half-life is the time it takes for half of the material to decay.
Example
A certain drug that a patient must receive has a half life of 13 days. If it is no longer potent once it has reached 40% of its original mass, how long after production does the doctor have to administer the drug?