# Exponential Growth & Decay

**Exponential ****growth and decay **is where a function’s growth or decay rate is proportional to the function’s current value. Firstly, we can see exponential growth in things like an increasing population. That is, population size grows at a rate proportional to the number currently in the population. Secondly, exponential decay occurs when the decay rate is proportional to the function’s current value. For example, we can see this in radioactive decay or drug concentration in a bloodstream.

In a continuous setting, Â we can use to model exponential growth, whereas we should use to model exponential decay. Note that mathematical modelling has drawbacks. That is, models often need refinements and fail when populations become unpredictable. See the problems and pitfalls of mathematically modelling the 2020 Coronavirus spread.

See Example 1 for more on Exponential Growth and Example 2 for Exponential Decay.

## Geometric Growth & Decay

**Geometric growth and decay **is the same as exponential growth and decay except the function is only evaluated at discrete values. For example, the geometric series with a start value of 5 and a common ratio of 2, i.e. is an example of a series that exhibits exponential growth discretely. This is geometric growth. However, the series is an example of a series that exhibits exponential decay discretely. This is geometric decay.

Compound interest is an example of geometric growth that you have seen before. Compound interest is where we earn interest on interest. That is, if a savings account has a an interest rate of 5% per annum and Â£2000 is deposited into the savings account, after one year there will be . Then there will be after two years. Generalising, there will be after n years.

At this level, you will mostly study exponential growth and decay.

## Examples

## Videos

Using simultaneous equations to find (and then interpreting) the unknown constants that determines the exponential growth in the value of a vintage car.