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Exponential growth is a type of growth where the rate of change is proportional to its quantity. In this growth, the value increases more rapidly over time.
The exponential growth function is written as:
where
Where:
Exponential growth is often found in real life, such as the growth of bacteria that divide into two at regular time intervals.
A researcher observes bacterial growth on a host with an initial condition of . These bacteria divide into two every .
If is the growth phase of bacteria every , then:
| Phase () | ||||||
|---|---|---|---|---|---|---|
| Number of bacteria |
It can be observed that:
From this pattern, bacterial growth can be modeled with an exponential function:
The graph of the function shows growth that gets faster as the value of increases. A characteristic feature of exponential growth graphs is their increasingly steep curve.
If we want to calculate the number of bacteria at hour , we need to know that hour occurs at phase (because each phase is ):
So, at hour , there are .
Exponential growth can have different initial values. In general, if the initial value is , then the exponential growth function becomes:
For example:
Sometimes we need to determine the initial value of an exponential growth if we know its value at a certain time.
Example case:
Suppose we know that the number of bacteria at phase is and the bacteria divide into two at each time interval. What is the initial number of bacteria?
We can use the equation with :
So, the initial number of bacteria is .
To calculate the number of bacteria after a longer time, we still use the same model. For example, to calculate the number of bacteria after :
Substitute the values and :
So, the number of bacteria after is .
E.coli bacteria cause diarrhea in humans. A researcher observed the growth of of these bacteria on a piece of food and found that these bacteria divide into every quarter of an hour.
Draw a table and graph showing the growth of these bacteria from phase to phase .
Model the function that describes the growth of E.coli bacteria every quarter of an hour.
Predict how many bacteria will be present after the first and .
In , positive HIV-AIDS cases totaled about . This number increased by an average of each year from to . If the increase in positive HIV cases in subsequent years is predicted to increase exponentially at each year, how many cases will occur in ?
E.coli bacteria with an initial count of that divide into two every .
Bacterial Growth:
| Phase () | ||||||
|---|---|---|---|---|---|---|
| Number of bacteria |
The E.coli bacterial growth function can be modeled as:
After the first means phase (bacteria divide every ):
HIV-AIDS cases:
The number of cases in was with a annual increase.
Mathematical model:
Exponential growth is also found in other contexts such as:
Exponential growth is very important in various fields such as biology, finance, and physics because it describes many real-world phenomena.
After the first means phase (bacteria divide every ):
To calculate cases in ( after ):
Therefore, the predicted number of HIV-AIDS cases in is approximately .
