Chapter 4.5, Problem 10E

(a)

To determine

To find : an approximation for the function $f\left(x\right)$ for values of $x$ near zero.

Answer to Problem 10E

  $f\left(x\right)\approx 4^{1/3}\left(1+\frac{x}{4}\right)$

Explanation of Solution

Given information :

  $f\left(x\right)={\left(4+3x\right)}^{1/3}$

Formula needed :

Use linear approximation, ${\left(1+x\right)}^k\approx 1+kx$ .

Take common $4$ from the function $f\left(x\right)={\left(4+3x\right)}^{1/3}$ .

  $\begin{array}{c}f\left(x\right)={\left(4\left(1+\frac{3}{4}x\right)\right)}^{1/3}\\ =4^{1/3}{\left(1+\frac{3}{4}x\right)}^{1/3}\end{array}$

Use linear approximation, $x$ as $\frac{3}{4}x$ and, $k=1/3$ .

  $\begin{array}{c}f\left(x\right)\approx 4^{1/3}\left(1+\frac{1}{3}\cdot \frac{3}{4}x\right)\\ =4^{1/3}\left(1+\frac{x}{4}\right)\end{array}$

(b)

To determine

To find : an approximation for the function $f\left(x\right)$ for values of $x$ near zero.

Answer to Problem 10E

  $f\left(x\right)\approx \sqrt{2}\left(1+\frac{x^2}{4}\right)$

Explanation of Solution

Given information :

  $f\left(x\right)=\sqrt{2+x^2}$

Formula needed :

Use linear approximation, ${\left(1+x\right)}^k\approx 1+kx$ .

Take common $2$ from the function $f\left(x\right)=\sqrt{2+x^2}$ .

  $\begin{array}{c}f\left(x\right)=\sqrt{2\left(1+\frac{x^2}{2}\right)}\\ =\sqrt{2}\cdot \sqrt{1+\frac{x^2}{2}}\\ =\sqrt{2}\cdot {\left(1+\frac{x^2}{2}\right)}^{1/2}\end{array}$

Use linear approximation, $x$ as $\frac{x^2}{2}$ and, $k=1/2$ .

  $\begin{array}{c}f\left(x\right)\approx \sqrt{2}\left(1+\frac{1}{2}\cdot \frac{x^2}{2}\right)\\ =\sqrt{2}\left(1+\frac{x^2}{4}\right)\end{array}$

(c)

To determine

To find : an approximation for the function $f\left(x\right)$ for values of $x$ near zero.

Answer to Problem 10E

  $f\left(x\right)\approx 1-\frac{2}{3\left(2+x\right)}$

Explanation of Solution

Given information :

  $f\left(x\right)=\sqrt[3]{{\left(1-\frac{1}{2+x}\right)}^2}$

Formula needed :

Use linear approximation, ${\left(1+x\right)}^k\approx 1+kx$ .

Simplify the power $f\left(x\right)=\sqrt[3]{{\left(1-\frac{1}{2+x}\right)}^2}$ .

  $\begin{array}{c}f\left(x\right)={\left(1-\frac{1}{2+x}\right)}^{2\cdot }{}^{1/3}\\ ={\left(1+\left(\frac{-1}{2+x}\right)\right)}^{2/3}\end{array}$

Use linear approximation, $x$ as $\frac{-1}{2+x}$ and, $k=2/3$ .

  $\begin{array}{c}f\left(x\right)\approx 1+\frac{2}{3}\cdot \left(\frac{-1}{2+x}\right)\\ =1-\frac{2}{3\left(2+x\right)}\end{array}$