Chapter 4, Problem 11PS

a.

To determine

Graph the functions by using graphing utility.

Answer to Problem 11PS

  

  

Explanation of Solution

Given information:

Graphical Reasoning

Use a graphing utility to graph the functions:

  $f\left(x\right)=2\cos 2x+3\sin 3x$

And

  $g\left(x\right)=2\cos 2x+3\sin 4x$

Calculation:

Here we are two functions:

  $f\left(x\right)=2\cos 2x+3\sin 3x$

  $g\left(x\right)=2\cos 2x+3\sin 4x$

Now, the graph for function $f\left(x\right)$ is:

  

Now, the graph for function $g\left(x\right)$ is:

  

Hence, the results are shown in the above graphs.

b.

To determine

Find the period of each function.

Answer to Problem 11PS

  $\boxed{f\left(x\right)=5\pi }$ , $\boxed{g\left(x\right)=3\pi }$ .

Explanation of Solution

Given information:

Graphical Reasoning

Use the graphs from part (a) to find the period of each function.

Calculation:

Hence, from the above graph it is cleared that the period of the function $f\left(x\right)$ and $g\left(x\right)$ is:

  $\boxed{f\left(x\right)=5\pi }$ , $\boxed{g\left(x\right)=3\pi }$ .

c.

To determine

Find if the function is periodic or not.

Answer to Problem 11PS

  $\boxed{Yes,itsperiodic}$ .

Explanation of Solution

Given information:

Graphical Reasoning

Is the function $h\left(x\right)=A\cos \alpha x+B\sin \beta x$ , where $\alpha$ and $\beta$ are positive integers, periodic? Explain.

Calculation:

Here, we are given the function:

  $h\left(x\right)=A\cos \alpha x+B\sin \beta x$

As, $\alpha$ and $\beta$ are positive integers.

We know that cosine function and sine function are periodic in nature.

Hence, $h\left(x\right)$ is also periodic as addition of two periodic functions is always periodic in nature.