Chapter 2.2, Problem 38E

$\left(a\right)$

To determine

To find: Why neither function is differentiable at $x=0$ .

Explanation of Solution

Given information: Recall that the numerical derivative $\left(\text{NDER}\right)$ can give meaningless values at points where function is not differentiable. In this exercise, we consider the numerical derivatives of the function $\frac{1}{x}$ and $\frac{1}{x^2}$ at $x=0$ .

The corresponding derivatives of the functions $\frac{1}{x}$ and $\frac{1}{x^2}$ are as follows:

  $-\frac{1}{x^2}$ and $-\frac{2}{x^3}$

The above numerical derivatives tend to infinity and become undefined at $x\to 0$ .

Hence, neither function is differentiable at $x=0$ .

$\left(b\right)$

To determine

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

Answer to Problem 38E

Undefined

Explanation of Solution

Given information: Recall that the numerical derivative $\left(\text{NDER}\right)$ can give meaningless values at points where function is not differentiable. In this exercise, we consider the numerical derivatives of the function $\frac{1}{x}$ and $\frac{1}{x^2}$ at $x=0$ .

Since NDER for both the functions $\frac{1}{x}$ and $\frac{1}{x^2}$ at $x=0$ tend to have a very large value. They will be undefined.

$\left(c\right)$

To determine

To find: Why NDER returns wrong responses that are so different from each other for these two functions.

Explanation of Solution

Given information: By analyzing the definition of the symmetric difference quotient, explain why NDER returns wrong responses that are so different from each other for these two functions.

NDER returns wrong responses that are so different from each other for these two functions as the numerical derivatives of the functions tends to have a very large value of order ${10}^8$ which can be taken as infinity.