Chapter 5.3, Problem 83E

a.

To determine

From the graph, determine whether the function is periodic; if it is periodic, find the period.

  $y=\left|\sin x\right|$

Answer to Problem 83E

  

The function is clearly periodic with period $\pi$ .

Explanation of Solution

Given information:

Use a graphing device to graph the following functions.

Calculation:

Periodic functions are functions that repeat themselves.

The function $f$ is periodic if $f\left(x\right)=f\left(t+p\right)$ , where $p$ is function’s period.

Using this definition, we will graph the function below,

  

Hence, the function is clearly periodic with period $\pi$ .

b.

To determine

From the graph, determine whether the function is periodic; if it is periodic, find the period.

  $y=\sin \left|x\right|$

Answer to Problem 83E

  

The function is not periodic.

Explanation of Solution

Given information:

Use a graphing device to graph the following functions.

Calculation:

Periodic functions are functions that repeat themselves.

The function $f$ is periodic if $f\left(x\right)=f\left(t+p\right)$ , where $p$ is function’s period.

Using this definition, we will graph the function below,

  

The interval between $-\pi$ and $\pi$ is not repeated .

Hence, the function is not periodic.

c.

To determine

From the graph, determine whether the function is periodic; if it is periodic, find the period.

  $y=2^{\cos x}$

Answer to Problem 83E

  

The function is clearly periodic with period $2\pi$ .

Explanation of Solution

Given information:

Use a graphing device to graph the following functions.

Calculation:

Periodic functions are functions that repeat themselves.

The function $f$ is periodic if $f\left(x\right)=f\left(t+p\right)$ , where $p$ is function’s period.

Using this definition, we will graph the function below,

  

Hence, the function is clearly periodic with period $2\pi$ .

d.

To determine

From the graph, determine whether the function is periodic; if it is periodic, find the period.

  $y=x-\Vert x\Vert$

Answer to Problem 83E

This function is periodic with period $1$ .

Explanation of Solution

Given information:

Use a graphing device to graph the following functions.

Calculation:

This graph of $y=x-\Vert x\Vert$ can easily be created on any graphing calculator that has a greater integer function, usually called $INT$ .

The graph is an infinite number of parallel straight lines with a slope of $1$ .

This function is not continuous.

Each line begins at $y=0$ and ends with an open circle at $y=1$ .

This indicates that the point is not the part of the graph.

Hence, this function is periodic with period $1$ .

e.

To determine

From the graph, determine whether the function is periodic; if it is periodic, find the period.

  $y=\cos \left(\sin x\right)$

Answer to Problem 83E

  

The function is clearly periodic with period $\pi$ .

Explanation of Solution

Given information:

Use a graphing device o graph the following functions.

Calculation:

Periodic functions are functions that repeat themselves.

The function $f$ is periodic if $f\left(x\right)=f\left(t+p\right)$ , where $p$ is function’s period.

Using this definition, we will graph the function below,

  

Hence, the function is clearly periodic with period $\pi$ .