Exponential Growth

Definition of Exponential Growth Function

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:

$f(x) = a^x$ where $a > 1$

Where:

• $a$ is the base (growth constant)
• $x$ is the variable (time)

Examples of Exponential Growth

Exponential growth is often found in real life, such as the growth of bacteria that divide into two at regular time intervals.

Bacterial Growth

A researcher observes bacterial growth on a host with an initial condition of $30 \text{ bacteria}$. These bacteria divide into two every $30 \text{ minutes}$.

If $x$ is the growth phase of bacteria every $30 \text{ minutes}$, then:

It can be observed that:

• For $x = 0$, the number of bacteria is $30$
• For $x = 1$, the number of bacteria is $60$
• For $x = 2$, the number of bacteria is $120 = 2^2 \cdot 30$
• For $x = 3$, the number of bacteria is $240 = 2^3 \cdot 30$
• For $x = 4$, the number of bacteria is $480 = 2^4 \cdot 30$

From this pattern, bacterial growth can be modeled with an exponential function:

Visualization of Exponential Growth Graph

The graph of the function $f(x) = 30 \cdot (2^x)$ shows growth that gets faster as the value of $x$ increases. A characteristic feature of exponential growth graphs is their increasingly steep curve.

Calculating the Number of Bacteria at a Specific Time

If we want to calculate the number of bacteria at hour $5$, we need to know that hour $5$ occurs at phase $10$ (because each phase is $30 \text{ minutes}$):

So, at hour $5$, there are $30{,}720 \text{ bacteria}$.

Variations in Initial Value

Exponential growth can have different initial values. In general, if the initial value is $P_0$, then the exponential growth function becomes:

For example:

• If the initial number of bacteria is $50$, then $f(x) = 50 \cdot (2^x)$
• If the initial number of bacteria is $100$, then $f(x) = 100 \cdot (2^x)$
• If the initial number of bacteria is $200$, then $f(x) = 200 \cdot (2^x)$

Determining the Initial Value

Sometimes we need to determine the initial value $x_0$ 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 $2$ is $8{,}000$ and the bacteria divide into two at each time interval. What is the initial number of bacteria?

We can use the equation $x_2 = a^2 \cdot x_0$ with $a = 2$:

So, the initial number of bacteria is $2{,}000 \text{ bacteria}$.

Calculating Long-Term Growth

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 $10 \text{ hours}$:

Substitute the values $a = 2$ and $x_0 = 2{,}000$:

So, the number of bacteria after $10 \text{ hours}$ is $2{,}048{,}000 \text{ bacteria}$.

Exercises

1. E.coli bacteria cause diarrhea in humans. A researcher observed the growth of $50$ of these bacteria on a piece of food and found that these bacteria divide into $2$ every quarter of an hour.

   1. Draw a table and graph showing the growth of these bacteria from phase $0$ to phase $5$.

   2. Model the function that describes the growth of E.coli bacteria every quarter of an hour.

   3. Predict how many bacteria will be present after the first $3 \text{ hours}$ and $4 \text{ hours}$.

2. In $2015$, positive HIV-AIDS cases totaled about $36 \text{ million people}$. This number increased by an average of $2\%$ each year from $2010$ to $2015$. If the increase in positive HIV cases in subsequent years is predicted to increase exponentially at $2\%$ each year, how many cases will occur in $2020$?

Answer Key

1. E.coli bacteria with an initial count of $50 \text{ bacteria}$ that divide into two every $15 \text{ minutes}$.

   1. Bacterial Growth:

      Phase ($15 \text{ minutes}$)	$0$	$1$	$2$	$3$	$4$	$5$
      Number of bacteria	$50$	$100$	$200$	$400$	$800$	$1{,}600$

      Bacterial Growth

      Growth of bacteria that divide into two every $15 \text{ minutes}$.

   2. The E.coli bacterial growth function can be modeled as:

      $$f(x) = 50 \cdot (2^x)$$

   3. After the first $3 \text{ hours}$ means phase $12$ (bacteria divide every $15 \text{ minutes}$):

      $$f(12) = 50 \cdot (2^{12})$$

      $$f(12) = 50 \cdot 4{,}096$$

      $$f(12) = 204{,}800$$

      After the first $4 \text{ hours}$ means phase $16$ (bacteria divide every $15 \text{ minutes}$):

      $$f(16) = 50 \cdot (2^{16})$$

      $$f(16) = 50 \cdot 65{,}536$$

      $$f(16) = 3{,}276{,}800$$

2. HIV-AIDS cases:

   The number of cases in $2015$ was $36{,}000{,}000 \text{ people}$ with a $2\%$ annual increase.

   Mathematical model:

   $$f(x) = 36{,}000{,}000 \cdot (1 + 0.02)^x$$

   To calculate cases in $2020$ ($5 \text{ years}$ after $2015$):

   $$f(5) = 36{,}000{,}000 \cdot (1.02)^5$$

   $$f(5) = 36{,}000{,}000 \cdot 1.1040808$$

   $$f(5) = 39{,}746{,}908$$

   Therefore, the predicted number of HIV-AIDS cases in $2020$ is approximately $39{,}746{,}908 \text{ people}$.

Other Applications of Exponential Growth

Exponential growth is also found in other contexts such as:

• Compound interest in investments
• Population growth of living organisms
• Radioactive decay (negative growth)
• Spread of infectious diseases

Exponential growth is very important in various fields such as biology, finance, and physics because it describes many real-world phenomena.