A cube root by tangent line

MediumCalculus~3m

Without a calculator, estimate $\sqrt[3]{28}$ using a tangent-line (linear) approximation of a well-chosen function at a nearby point where you know the value exactly. Give your estimate to four decimal places.

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Linearize $f(x)=\sqrt[3]{x}$ about the nearest perfect cube $a=27$ : $f(x)\approx f(a)+f'(a)(x-a)$ .
$f(27)=3$ and $f'(27)=\tfrac13\cdot 27^{-2/3}=\tfrac1{27}$ ; evaluate at $x=28$ .

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3.037 (± 0.001)

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A cube root by tangent line

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