Chapter 11.4, Problem 51E

Solution

To explain: Why NDER of $f\left(x\right)=\left|x\right|$ evaluated at $x=0$ is equal to $0$ .

The reason why many calculators display incorrect answer for NDER of $f\left(x\right)=\left|x\right|$ evaluated at $x=0$ is explained.

Given information:

The outcome for NDE of $f\left(x\right)=\left|x\right|$ evaluated at $x=0$ of many calculators is given.

Calculation:

At $x=0$ for $f\left(x\right)=\left|x\right|$ , a sharp corner can be observed.

And that is why, there cannot be any tangent and consequently no derivative at this location.

The incorrect value is displayed on the calculator because it considers $f\left(0+h\right)$ and $f\left(0-h\right)$ as equal. As a result, it finds the numerical derivative $f\left(h\right)=f\left(-h\right)$ for $f\left(x\right)=\left|x\right|$ .