Chapter 11.1, Problem 38E

Solution

a)

To find:the graph of the function.

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

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin x}{x}\text{ if }x\ne 0\\ 1\text{ if }x=0\end{array}\right.$ at $x=0$ .

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 of the function $f$ at $x=0$ is 0.

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin x}{x}\text{ if }x\ne 0\\ 1\text{ if }x=0\end{array}\right.$ at $x=0$ and the graph of the function is shown in Figure (1).

Formula used:

Calculation:

Derivative of a function $f$ at $x=0$:

By observing Figure (1), the tangent of the function $f$ at $x=0$ is $y=1$ . The slope of the tangent is 0.

Thus, the derivative of $f$ at $x=0$ is 0.

c)

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

The function is differentiable at $x=0$ .

Given information:

The function is $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin x}{x}\text{ if }x\ne 0\\ 1\text{ if }x=0\end{array}\right.$ at $x=0$ and the graph of the function is shown in Figure (1).

Calculation:

The function is differentiable at $x=2.$