Chapter 4, Problem 1CRE

To determine

Estimate using the Linear Approximation or linearization. Also, use a calculator to estimate the error.

Answer to Problem 1CRE

Solution:

The error is $3.448\times {10}^{-5}$.

Explanation of Solution

Given:

The expression is ${8.1}^{\frac{1}{3}}-2$.

Formula used:

Linear Approximation:

$\Delta f=f^{\prime }\left(a\right)\Delta x$

Calculation:

The expression is ${8.1}^{\frac{1}{3}}-2$.

Let $f\left(x\right)=x^{\frac{1}{3}}$, $a=8\text{and}\Delta x=0.1$.

Now, find the derivative of the function $f\left(x\right)$ with respect to $x$.

$f^{\prime }\left(x\right)=\frac{1}{3}{x}^{-\frac{2}{3}}$

Find the value of $f^{\prime }\left(a\right)$.

$\begin{array}{c}{f}^{\prime }\left(a\right)=\frac{1}{3}{a}^{-\frac{2}{3}}\\ =\frac{1}{3}{\left(8\right)}^{-\frac{2}{3}}\\ =\frac{1}{3}{\left(2^3\right)}^{-\frac{2}{3}}\\ =\frac{1}{3}\left(\frac{1}{4}\right)\\ =\frac{1}{12}\end{array}$

Now, find the estimate using Linear Approximation.

Use the formula: $\Delta f=f^{\prime }\left(a\right)\Delta x$

$\begin{array}{c}\Delta f=f^{\prime }\left(a\right)\Delta x\\ =\frac{1}{12}\left(0.1\right)\\ =0.00833333333\end{array}$

Use the calculator to find the value of ${8.1}^{\frac{1}{3}}-2$.

We can rewrite it as:

${\left(8.1\right)}^{0.33333}-2$

$\begin{array}{c}{\left(8.1\right)}^{0.33333}-2=2.00829885-2\\ =0.00829885\end{array}$

Now, find the error in the Linear Approximation.

$\begin{array}{c}\text{Error}=\left|0.00829885-0.00833333333\right|\\ =0.00003448\\ =3.448\times {10}^{-5}\end{array}$

Hence, the error is $3.448\times {10}^{-5}$.

Conclusion:

With the help of the formula, wecould figure out the error in the Linear Approximation.