Chapter 13.5, Problem 8PPSE

Solution

To find: The period and amplitude of cosine function and the values of $\left(x\right)$ for $0\le x\le 2\pi$ where the maximum, minimum and zeros occur as shown in graph below.

  

The period of the function is $\frac{2\pi }{3}$ and its amplitude is $2$ . And for the interval $0\le x\le 2\pi$ , the maximum value occurs at $x=0,\frac{2\pi }{3},\frac{4\pi }{3}\text{ and }2\pi$ , the minimum value occurs at $x=\frac{\pi }{3},\pi \text{ and }\frac{5\pi }{3}$ and the zeros occur at $x=\frac{\pi }{6},\frac{\pi }{2}\text{,}\frac{5\pi }{6}\text{,}\frac{7\pi }{6},\frac{3\pi }{2}\text{ and }x=\frac{11\pi }{6}$ .

Given:

The graph of the cosine function is shown below.

  

Calculation:

From the graph of the cosine function and the given parameters as shown in graph, it can be seen that the maximum value of the function is $2$ and minimum value of the function is $-2$ .

Therefore, amplitude $\left(a\right)$ of the function is evaluated as:

  $\begin{array}{c}a=\frac{1}{2}\left(2-\left(-2\right)\right)\\ =2\end{array}$

In the interval from $0\text{ to }2\pi$ , there exists $3$ cycle. Therefore, the period $\left(T\right)$ of the function is evaluated as:

  $\begin{array}{c}\text{Period}\left(T\right)=\frac{2\pi }{b}\\ =\frac{2\pi }{3}\end{array}$

And in the interval from $0\text{ to }2\pi$ , the maximum value of the curve is $2$ which occurs at $x=0,\frac{2\pi }{3},\frac{4\pi }{3}\text{ and }2\pi$ .

Similarly, in the interval from $0\text{ to }2\pi$ , the minimum value of the curve is $-2$ which occurs at $x=\frac{\pi }{3},\pi \text{ and }\frac{5\pi }{3}$ .

Now, from the graph of cosine curve it can be seen that in the interval from $0\text{ to }2\pi$ , the curve cuts $x-$ axis at points $x=\frac{\pi }{6},\frac{\pi }{2}\text{,}\frac{5\pi }{6}\text{,}\frac{7\pi }{6},\frac{3\pi }{2}\text{ and }x=\frac{11\pi }{6}$ . Therefore, the zeros of the cosine curve occur at $x=\frac{\pi }{6},\frac{\pi }{2}\text{,}\frac{5\pi }{6}\text{,}\frac{7\pi }{6},\frac{3\pi }{2}\text{ and }x=\frac{11\pi }{6}$ .

Conclusion:

The period of the function is $\frac{2\pi }{3}$ and its amplitude is $2$ . And for the interval $0\le x\le 2\pi$ , the maximum value occurs at $x=0,\frac{2\pi }{3},\frac{4\pi }{3}\text{ and }2\pi$ , the minimum value occurs at $x=\frac{\pi }{3},\pi \text{ and }\frac{5\pi }{3}$ and the zeros occur at $x=\frac{\pi }{6},\frac{\pi }{2}\text{,}\frac{5\pi }{6}\text{,}\frac{7\pi }{6},\frac{3\pi }{2}\text{ and }x=\frac{11\pi }{6}$ .