Chapter 3.9, Problem 14E

To determine

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

Answer to Problem 14E

The values of x for which the linear approximation is accurate within $0.1$ are lies in $-0.483<x<0.416$ .

Explanation of Solution

Given information:

The given function is,

  $f\left(x\right)=e^x$

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,

  $\begin{array}{c}L\left(x\right)=f\left(a\right)+f^{\prime }\left(a\right)\left(x-a\right)\\ =\left(e^a\right)+\left(e^a\right)\left(x-a\right)\\ =\left(e^0\right)+\left(e^0\right)\left(x-0\right)\\ =1+x\end{array}$

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 $-0.483<x<0.416$ .