Exponential Growth and Decay

Exponential growth is where a function’s growth rate is proportional to the function’s current value. Exponential growth can be seen in things like human population or virus spreading. Exponential decay occurs when the decay rate is proportional to the function’s current value, i.e. the function is decreasing in a similar fashion instead of increasing.

Geometric growth or decay is the same as exponential growth or decay except the function is only evaluated at discrete values. Example, recall a geometric series that has a start value of 5 and a common ratio of 2, i.e. $5+10+20+40+80+...$ is an example of a series that exhibits geometric growth. The series $10+1+0.1+0.01+0.001+...$ is an example of a series that exhibits geometric decay.

Example 1 – Compound interest is an example of a geometric series that you have seen before. Compound interest is where interest is earned on interest. If a savings account has a an interest rate of 5% per annum and £2000 is deposited into the savings account, after one year there will be $2000\times 1.05=2100$. After two years there will be $2000\times 1.05^2=2205$. After n years there will be $2000\times 1.05^n$.

Example 2 – A rare species of dandelion is being observed. The population, p, after a given number of years, T, is given by

$p(t)=\frac{500e^{0.3t}}{1+4e^{0.3t}}$

a) Find the population of dandelion at the beginning of the study.
b) Find the rate at which the dandelion population changes after 5 years. Give your answer to two decimal places.
c) Explain why the population of dandelion does not exceed 125.