Question

Find the linear approximation of the function
f(x)=^3root(1+x)
at a=0 and use it to approximate ^3root(0.95) and ^3root(1.01).

Answer 1

\[f(x)=\sqrt[3]{1+x}\]
\[\sqrt[3]{0.95} \]
\[\sqrt[3]{1.01}\]

Answer 2

\[\sqrt[3]{1+x}\] has derivative
\[\frac{1}{3\sqrt[3]{x+1}^2}\]
at \(x=0\) you get the slope of \(\frac{1}{3}\) so the equation for your line is
\[y=\frac{1}{3}x+1\]

Answer 3

that is the "linearization" at \(x=0\)
for \(1.01\) use \(x=.01\) and for \(.95\) use \(x=-.05\)