Chapter 3.2, Problem 38E

(a)

To determine

To find: The reason why the neither function is differentiable at $x=0$ .

Answer to Problem 38E

The function is differentiable at $x=0$ as at $x=0$ is not their domains of the functions $\frac{1}{x}$ and $\frac{1}{x^2}$ , they are both discontinuous at x=0.

Explanation of Solution

Given Data:

The numerical derivative of the function $\frac{1}{x}$ at $\frac{1}{x^2}=0$ .

Calculation:

Consider the given function are,

  $\begin{array}{l}\frac{1}{x}=0\\ \frac{1}{x^2}=0\end{array}$

Since at $x=0$ is not their domains of the functions $\frac{1}{x}$ and $\frac{1}{x^2}$ , they are both discontinuous at x=0.

Thus, the function is differentiable at $x=0$ .

(b)

To determine

To find: The NDER at each function for $x=0$ .

Answer to Problem 38E

The value of $NDER$ for $\frac{1}{x^2}$ is 0 and for $\frac{1}{x}=1000000$ .

Explanation of Solution

Consider that for $\frac{1}{x}$ ,

  $\begin{array}{c}\text{NDER}\left(\frac{1}{x},0\right)=\frac{f\left(0+0.001\right)-f\left(0-0.001\right)}{2\left(0.001\right)}\\ =\frac{\frac{1}{0.001}-\left(\frac{1}{-0.001}\right)}{0.002}\\ =1000000\end{array}$

Consider that for $\frac{1}{x^2}$ ,

  $\begin{array}{c}\text{NDER}\left(\frac{1}{x^2},0\right)=\frac{f\left(0+0.001\right)-f\left(0-0.001\right)}{2\left(0.001\right)}\\ =\frac{{\left(\frac{1}{0.001}\right)}^2-{\left(\frac{1}{-0.001}\right)}^2}{0.002}\\ =0\end{array}$

(c)

To determine

To find: The reason for why NDER returns wrong responses that are so different from each other for these two function.

Answer to Problem 38E

The reason is that the responses differ from each other due to $\frac{1}{x^2}$ is even that determines that $\text{NDER}\left(\frac{1}{x^2},0\right)=0$ where $\frac{1}{x}$ is odd.

Explanation of Solution

For the incorrect returns because of the even through these functions are not defined at $x=0$ as they are defined at $x=\pm 0.001$ .

Since, the responses differ from each other due to $\frac{1}{x^2}$ is even that determines that $\text{NDER}\left(\frac{1}{x^2},0\right)=0$ where $\frac{1}{x}$ is odd.