Chapter 11.1, Problem 37E

Solution

a)

To find: The graph of the function.

The graph of the function is shown in Figure (1).

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}\frac{\left|x-2\right|}{x-2}\text{ if }x\ne 2\\ 1\text{ if }x=2\end{array}\right.$ at $x=2$ .

Calculation:

The graph of the function is shown in Figure (1).

  

Figure (1)

Therefore, the graph of the function is shown above.

b)

To find:the derivative of the function at the given point if it exists.

The derivative doesn’t exist at a given point.

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}\frac{\left|x-2\right|}{x-2}\text{ if }x\ne 2\\ 1\text{ if }x=2\end{array}\right.$ at $x=2$ and the graph of the function is shown in Figure (1).

Calculation:

The graph of the function does not have a definable slope at $x=2$ , the derivative of $f$ does not exist at $x=2$ .

c)

To find:the reason if the derivative does not exist at the point.

The function is not continuous at the point $x=2$ .

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}\frac{\left|x-2\right|}{x-2}\text{ if }x\ne 2\\ 1\text{ if }x=2\end{array}\right.$ at $x=2$ and the graph of the function is shown in Figure (1).

Calculation:

Note that there is a break in the graph of the function $f$ at $x=2$ . That is the function $f$ is not continuous at $x=2$ . The derivative does not exist at points where functions have discontinuous.