Use linear approximation to estimate $sqrt[3]8.2$ correct to three decimal places. Write $f(x)=x^{1/3}$ and take base point $a=8$.

Step-by-step solution

1. Set $f(x)=x^{1/3}$ so $f(a)=\sqrt[3]{8}=2$. Compute derivative $f^{\prime }(x)=\frac{1}{3}{x}^{-2/3}=\frac{1}{3x^{2/3}}$. Evaluate at $a$: $f^{\prime }(8)=\frac{1}{3\cdot 8^{2/3}}=\frac{1}{3\cdot 4}=1/12\approx 0.08333$. Difference $\Delta x=8.2-8=0.2$. Linear approximation: $f(8.2)\approx f(8)+f^{\prime }(8)\Delta x=2+0.08333\times 0.2=2.016666$. Round to three decimals: $2.017$.
2. final answer
3. :
4. 2.017

Concept tested: Linear approximation (tangent line estimate)

Published 7 October 2026 · How this solution was prepared and checked