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Section 2.5 Relative change and elasticity
Objectives
Use differentials to construct linear approximations near a point.
Interpret a function’s linearization as its best local straight-line model.
Evaluate accuracy using concavity and basic error estimates.
Relative differential:
\(\rd{y}=\dfrac{\dd{y}}{y}\)
Can be interpreted in terms of percent change:
\(\dfrac{\dd{y}}{y}\cdot 100\%\)
Relative derivative aka semi-elasticity:
\(\dfrac{\rd{y}}{x}=\dfrac{\dd{y}/\dd{x}}{y}\)
Elasticity of a function:
Aside The elasticity of a power function is its degree, and in the limit, the elasticity of a polynomial is its degree. This can in some sense be used to define the degree of non-power functions.
\begin{equation*}
E=\rdv{y}{x}=\frac{\dd{y}/y}{\dd{x}/x}=\frac{x}{y}\dv{y}{x}=\frac{\dd{y}/\dd{x}}{y/x}
\end{equation*}
This is useful in economics (e.g. price elasticity of demand). Maybe use this as the guiding application? Also apparently useful in chemistry when talking about reactions.
A function has constant elasticity if and only if it is a power function. That is, the elasticity of
\(x^a\) is
\(E=a\text{.}\)
A function is elastic if
\(|E|\gt 1\text{,}\) inelastic if
\(|E|\lt 1\text{,}\) and unit elastic if
\(|E|=1\text{.}\)
To find where revenue is maximized, we can find where the elasticity of demand is unit elastic. (Prove this by showing that the elasticity of
\(R(x)=xD(x)\) is
\(E_R=1+E_D\text{.}\) )
