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Section 5.2 Relative Rate of Change
Subsection 5.2.1 Relative Change and Percentage Change
Definition 5.2.1 .
The relative change of a quantity \(x\text{,}\) is the ratio,
\begin{equation*}
\boxed{\frac{\text{absolute change}}{\text{quantity}} = \frac{\Delta x}{x}}
\end{equation*}
The percentage change of \(x\) is the relative change of \(x\text{,}\) expressed as a percentage, or,
\begin{equation*}
\boxed{\text{percentage change in $x$} = \frac{\Delta x}{x} \times 100\%}
\end{equation*}
Subsection 5.2.2 Sensitivity to Change
Definition 5.2.2 .
If a small change in \(x\) produces a large change in \(f(x)\text{,}\) then we say that the function \(f\) is sensitive to changes in \(x\text{.}\)
In this way, the derivative \(f'(x)\) is a measure of the sensitivity of the \(f\) to changes in \(x\text{.}\)
Subsection 5.2.3 Relative Growth Rate and the Natural Logarithm
In economics and other fields, the relative growth of something is much more important than its absolute growth.
Definition 5.2.3 .
Let \(y = y(t)\) represent a quantity over time. Then, the relative rate of change of \(y\) is given by,
\begin{equation*}
\boxed{\text{RRC} = \frac{y'(t)}{y(t)}}
\end{equation*}
This gives RRC as a decimal, which can be written as a percentage by multiplying by 100. Notice that,
\begin{equation*}
\frac{y'(t)}{y(t)} = \frac{d}{dt} \ln{y(t)}
\end{equation*}
That is, the relative growth rate is the rate of change of the natural logarithm of the original outputs \(y(t)\text{.}\) This is another reason why logarithms with base \(e\) are considered natural.
