Using Linear Approximation, estimate $f$ for a change in $x$ from $x=a$ to $x=b.$

$f(x)=x^{\frac{1}{5}},a=32,b=32\mathrm{.}5$

Consider the work to estimate $f.$

$f^{\text{'}}(x)=\frac{1}{5}{x}^{-45}$

$f^{\text{'}}(a)=\frac{1}{5}*{32}^{-45}=\frac{1}{80}$

$f$~~$f^{\text{'}}(a)x=\frac{1}{80}(-0\mathrm{.}5)=-0\mathrm{.}00625$

Identify the first error in the estimate of $f.$

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