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Section 4.11 : Linear Approximations

3. Find the linear approximation to \(g\left( z \right) = \sqrt[4]{z}\) at \(z = 2\). Use the linear approximation to approximate the value of \(\sqrt[4]{3}\) and \(\sqrt[4]{{10}}\). Compare the approximated values to the exact values.

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We’ll need the derivative first as well as a couple of function evaluations.

\[g'\left( z \right) = \frac{1}{4}{z^{ - \,\frac{3}{4}}}\hspace{0.5in}g\left( 2 \right) = {2^{\frac{1}{4}}}\hspace{0.5in}g'\left( 2 \right) = \frac{1}{4}\left( {{2^{ - \,\frac{3}{4}}}} \right)\] Show Step 2

Here is the linear approximation.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{L\left( z \right) = {2^{\frac{1}{4}}} + \frac{1}{4}\left( {{2^{ - \,\frac{3}{4}}}} \right)\left( {z - 2} \right)}}\] Show Step 3

Finally, here are the approximations of the values along with the exact values.

\[\begin{align*}L\left( 3 \right) & = 1.33786 & \hspace{0.5in}g\left( 3 \right) & = 1.31607 & \hspace{0.5in} {\mbox{% Error : }} & 1.65523\\ L\left( {10} \right) & = 2.37841 & \hspace{0.5in}g\left( {10} \right) & = 1.77828 & \hspace{0.5in} {\mbox{% Error : }} & 33.7481\end{align*}\]

So, as we might have expected the farther from \(z = 2\) we got the worse the approximation is. Recall that the approximation will generally be more accurate the closer to the point of the linear approximation.