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Section 3.3 Intro to Exponential Growth and Decay

Many physical phenomena can be modelled using exponential functions. They are classified in two categories: exponential growth, and exponential decay. In general, exponential growth or decay occurs when a quantity grows or decays at a rate proportional to their size.

An example of exponential growth is population growth. For many organisms, their population increases at a rate proportional to their size. This is because the larger the population, the more people available to reproduce, so the more offspring are produced.

An example of exponential decay is radioactive decay, the behavior of radioactive isotopes as they decay into non-radioactive isotopes. The more atoms which are radioactive, the more of them can decay into non-radioactive isotopes.

Subsection 3.3.1 Exponential Change

Consider a quantity \(y\text{,}\) which currently has value \(y_0\text{,}\) and which is growing at a rate of \(r\) percent per year (here \(r\) is the percentage change, written as a decimal). Then, after one year, the new value of the quantity is its current value \(y_0\text{,}\) plus the change in the quantity. Since the quantity is growing by \(r\) \%, this means that the quantity grew by \(r \cdot y_0\text{.}\) Then,

\begin{align*} \text{new $y$} \amp = \text{old $y$} + \text{change in $y$}\\ y \amp = y_0 + r \cdot y_0 \end{align*}

Then, combining like terms,

\begin{equation*} y = (1 + r) y_0 \end{equation*}

The factor \(1 + r\) is multiplied by the current quantity in order to get the new quantity, and is called the growth factor (or decay factor, with exponential decay).