How do you know When to use an exponential growth model?

Question

Ezra Adams · Accepted Answer

When the growth is at an approximately constant percent per unit time, an exponential model works well.

For example, if a population is growing at 4% per year, with an initial population of 1000, then #P=f(t)=1000*1.04^t# is a good model. Note that #(f(t+1))/f(t)=1.04# for all #t#, meaning that there is 4% growth over every time interval of length 1. This can also be written in terms of #e approx 2.71828# as #P=f(t)=1000*e^(kt)#. The value of #k# is determined by setting #e^(k)=1.04# so that #k=ln(1.04) approx 0.0392#. This means that a 4% growth rate per unit time corresponds to approximately 3.92% instantaneous growth rate. This basically means that the tangent line at any point grows by approximately 3.92% over a time interval of length 1.

Adam Adkins · Answer

undefined You should use an exponential growth model when the rate of growth of a quantity is proportional to its current value, meaning the larger the quantity, the faster it grows. This pattern is characteristic of many natural phenomena such as population growth, compound interest, and the spread of diseases in the early stages. Additionally, exponential growth models are appropriate when there are no limiting factors that would cause the growth rate to slow down over time.