Chapter 8.4, Problem 4QR

To determine

To calculate: simplify the function.

Answer to Problem 4QR

Function $\sqrt{1+{\left(\frac{x}{4}-\frac{1}{x}\right)}^2}=\frac{x^2+4}{4x}$ .

Explanation of Solution

Given information: Function $\sqrt{1+{\left(\frac{x}{4}-\frac{1}{x}\right)}^2}\text{ on }\left[4,12\right]$ .

Formula used:

Algebraic formula

  ${\left(a+b\right)}^2=a^2+b^2+2ab$

Calculation:

Function $\sqrt{1+{\left(\frac{x}{4}-\frac{1}{x}\right)}^2}\text{ on }\left[4,12\right]$

  $\begin{array}{l}\sqrt{1+{\left(\frac{x}{4}-\frac{1}{x}\right)}^2}=\sqrt{1+{\left(\frac{x}{4}\right)}^2+{\left(\frac{1}{x}\right)}^2-\frac{1}{2}}\\ \sqrt{1+{\left(\frac{x}{4}-\frac{1}{x}\right)}^2}=\sqrt{\frac{{\left(x^2+4\right)}^2}{16x^2}}\\ \sqrt{1+{\left(\frac{x}{4}-\frac{1}{x}\right)}^2}=\frac{x^2+4}{4x}\end{array}$