Chapter 13.5, Problem 10PPSE

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 $2\pi$ and its amplitude is $4$ . And for the interval $0\le x\le 2\pi$ , the maximum value occurs at $x=\pi$ , the minimum value occurs at $x=0\text{ and }2\pi$ and the zeros occur at $x=\frac{\pi }{2}\text{ and }\frac{3\pi }{2}$ .

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 $4$ and minimum value of the function is $-4$ .

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

  $\begin{array}{c}a=\frac{1}{2}\left(4-\left(-4\right)\right)\\ =4\end{array}$

In the interval from $0\text{ to }2\pi$ , there exists $1$ cycle. Therefore, the period $\left(T\right)$ of the function is evaluated as:

  $\begin{array}{c}\text{Period}\left(T\right)=\frac{2\pi }{b}\\ =\frac{2\pi }{1}\\ =2\pi \end{array}$

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

Similarly, in the interval from $0\text{ to }2\pi$ , the minimum value of the curve is $-4$ which occurs at $x=0\text{ and }2\pi$ .

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 $x=\frac{\pi }{2}\text{ and }\frac{3\pi }{2}$ . Therefore, the zeros of the cosine curve occur at $x=\frac{\pi }{2}\text{ and }\frac{3\pi }{2}$ .

Conclusion:

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