Chapter 3.9, Problem 11E

To determine

To find: The values of x for which the linear approximation is accurate with in $0.1$ .

Answer to Problem 11E

The values of x for which the linear approximation is accurate within $0.1$ are lies in $-1.204<x<0.706$ .

Explanation of Solution

Given information:

The given function is,

  $f\left(x\right)=\sqrt[3]{1-x}$

The given value of $a$ is $0$ .

Concept used:

The Linearization of the function is calculated as,

  $L\left(x\right)=f\left(a\right)+f^{\prime }\left(a\right)\left(x-a\right)$

The Linearization of the function is calculated as,

  $\begin{array}{c}L\left(x\right)=f\left(a\right)+f^{\prime }\left(a\right)\left(x-a\right)\\ =\left(\sqrt[3]{1-a}\right)+\left(-\frac{1}{3}{\left(1-a\right)}^{-2/3}\right)\left(x-a\right)\\ =\left(\sqrt[3]{1-0}\right)+\left(\frac{1}{3}{\left(1-0\right)}^{-2/3}\right)\left(x-0\right)\\ =1-\frac{1}{3}x\end{array}$

Consider the graph of the function and the tangent line.

  

From the graph, the values of x for which the linear approximation is accurate within $0.1$ are lies in $-1.204<x<0.706$ .