Find the linearization of \( f(x)=(1+3 x)^{\frac{1}{4}} \quad \) at \( \quad x=2 \), then use this approximation to estimate the value of \( (7.003)^{1 / 4} \). (enter the exact answer (do not use decimals)) Solution. \begin{tabular}{|l|l|l|} \hline\( f^{\prime}(x) \) & \( = \) & \\ \hline \begin{tabular}{l} The linear approximation at \\ \( x=2 \) is \end{tabular} & \( L(x)= \) & \\ \hline\( (7.003)^{1 / 4} \) & \( \approx \) & \\ \hline \end{tabular}
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Using the chain rule, we get: \[f'(x) = \frac{1}{4}(1 + 3x)^{-\frac{3}{4}} \cdot 3 = \frac{3}{4}(1 + 3x)^{-\frac{3}{4}}\] Show more…
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