Chapter 4, Problem 14CT

To determine

Find the period of the sum of these functions is the least common factor of their individual periods

Answer to Problem 14CT

The period of the sum of these functions is the least common factor of their individual periods which is 2

Explanation of Solution

Given:

  $y=\sin 2\pi x+2\cos \pi x$

Consider the function:

  $y=\sin 2\pi x+2\cos \pi x$

Enter this equation into the graphing calculator as $Y_1$ . Press $\text{Y}=$ and enter the expression. The display is as shown below:

  

Press $[\text{ GRAPH }]$ to plot the graph of the above expression. The display is as shown below:

since, the sine and cosine functions are periodic; their sum will also be periodic.

The period of the function $\sin axis\frac{2\pi }{a}.$ Hence, the period of the function $\sin 2\pi x$ is:

  $\frac{2\pi }{2\pi }=1$

And, the period of the function $2\cos \pi x$ is:

  $\frac{2\pi }{\pi }=2$

The period of the sum of these functions is the least common factor of their individual periods which is 2. Hence, the period of the function is $2$