Chapter 4.6, Problem 11E

To determine

To sketch: the graph of the given function.

Answer to Problem 11E

The correct option is (a) and the period of the function is 1.

Explanation of Solution

Given information:

Given function is

  $y=\frac{1}{3}\cot \frac{\pi x}{2}$ .

  

Calculation:

We know that the vertical asymptotes for $y=\cot x$ are at $x=0$ and $x=\pi$ and its period is $\pi$ .

So for the given equation, let us we find out the values of the two consecutive asymptotes.

On comparing we get,

  $\begin{array}{c}\pi x=0\\ x=0\end{array}$

Also,

  $\begin{array}{c}\pi x=\pi \\ x=\frac{\pi }{\pi }\\ x=1\end{array}$

Therefore, the two consecutive asymptotes occur at $x=0$ and $x=1$ .

Next, we calculate the values of $y$ foe a few values of $x$ between these two asymptotes.

For example,

Substituting $x=0$

We get,

  $\begin{array}{c}y=\frac{1}{3}\cot \pi \left(0\right)\\ y=\frac{1}{2}\cot 0\\ y=\infty \end{array}$

Similarly, we calculate other values of $y$ and write them down in tabular form.

$x$	$0$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{6}$	$1$
$y$	Not defined	$\frac{1}{2}$	$0$	$\frac{\sqrt{3}}{2}$	Not defined

Now using these values, let us plot the graph from $-2$ to $2$ .

  

The above graph represents the equation $y=\frac{1}{2}\cot \left(\pi x\right)$ .

Now, as the above graph resembles the graph given in option (a).

Also, from the above graph we can see that the function $y=\frac{1}{2}\cot \left(\pi x\right)$ complete its onw cycle from 0 to 1. Thus, the period of the function is 1.

Therefore, the correct option is (a) and the period of the function is 1.