Solution Found!
?Find the linear approximation of the function \(g(x)=\sqrt[3]{1+x}\) at \(a=0\) and use
Chapter 2, Problem 6(choose chapter or problem)
Find the linear approximation of the function \(g(x)=\sqrt[3]{1+x}\) at \(a=0\) and use it to approximate the numbers \(\sqrt[3]{0.95}\) and \(\sqrt[3]{1.1}\). Illustrate by graphing \(t\) and the tangent line.
Questions & Answers
QUESTION:
Find the linear approximation of the function \(g(x)=\sqrt[3]{1+x}\) at \(a=0\) and use it to approximate the numbers \(\sqrt[3]{0.95}\) and \(\sqrt[3]{1.1}\). Illustrate by graphing \(t\) and the tangent line.
ANSWER:Step 1 of 5
Consider the given function.
\(\begin{aligned} g(x) & =\sqrt[3]{1+x} \\ & =(1+x)^{\frac{1}{3}} \end{aligned}\)
Differentiate the above function with respect to x.
\(\begin{aligned} g^{\prime}(x) & =\frac{1}{3}(1+x)^{\frac{1}{3}-1} \\ & =\frac{1}{3}(1+x)^{-\frac{2}{3}} \\ & =\frac{1}{3 \sqrt[3]{(1+x)^{2}}} \end{aligned}\)