Chapter 5.3, Problem 75E

a.

To determine

Is $f$ even, odd, or neither?

Answer to Problem 75E

Odd.

Explanation of Solution

Given information:

A function $f$ is given.

  $f\left(x\right)=\frac{1-\cos x}{x}$

Calculation:

Consider the function

  $f\left(x\right)=\frac{1-\cos x}{x}$

Now, sketch the graph of the function

  

First let’s decide on the function’s symmetry. It is clearly symmetric about the origin,

Hence it is odd.

b.

To determine

Find the $x$ -intercepts of the graph of $f$ .

Answer to Problem 75E

  $x=0,\pm 2\pi ,\pm 4\pi ,\pm 6\pi \mathrm{....}$

Explanation of Solution

Given information:

A function $f$ is given.

  $f\left(x\right)=\frac{1-\cos x}{x}$

Calculation:

Consider the function

  $f\left(x\right)=\frac{1-\cos x}{x}$

Now, sketch the graph of the function

  

The $x$ intercepts are the points where the graph touches the $x$ -axis, from the above graph we observe that the $x$ intercepts are occur at $x=0,\pm 2\pi ,\pm 4\pi ,\pm 6\pi \mathrm{....}$

c.

To determine

Graph $f$ in an appropriate viewing rectangle.

Answer to Problem 75E

  

Explanation of Solution

Given information:

A function $f$ is given.

  $f\left(x\right)=\frac{1-\cos x}{x}$

Calculation:

Consider the function

  $f\left(x\right)=\frac{1-\cos x}{x}$

Now, sketch the graph of the function

  

Sketch the graph

Start the graphing calculator

Make sure that the calculator is in radians mode

  

Press $\boxed{Y1=}$ key and enter the function as follows:

  $Y1=\boxed{1}\boxed{-}\boxed{\cos }\boxed{(}\boxed{X}\boxed{)}\boxed{)}\boxed{÷}\boxed{X},$ the display as shown below

  

Here to insert $\pi ,$ press $\boxed{2nd}+\boxed{{}^{\wedge }}$ now, we need to adjust the window settings as follows:

Press $\boxed{WINDOW}$ key then set

  $\boxed{\begin{array}{l}X\min =-20 and Y\min =-1 \\ X\max =20 Y\max =1\\ Xscl=4 Yscl=1 \end{array}}$

  

Press $\boxed{GRAPH}$ key to graph the function

  

Hence, the viewing rectangle $\left[-20,20\right]by\left[-1,1\right]$ shown in above figure gives a good global view of the graph of $f$ .

d.

To determine

Describe the behaviour of the function as $x\to \pm \infty$ .

Answer to Problem 75E

  $x\to \pm \infty ,f(x)\to 0$

Explanation of Solution

Given information:

A function $f$ is given.

  $f\left(x\right)=\frac{1-\cos x}{x}$

Calculation:

Consider the function

  $f\left(x\right)=\frac{1-\cos x}{x}$

Now, sketch the graph of the function

  

Hence, from the diminishing heights of the curve as $x$ gets very large or very small, we conclude that as $x\to \pm \infty ,f(x)\to 0$ .

e.

To determine

Notice that $f(x)$ is not defined when $x=0$ .

Answer to Problem 75E

  $x\to 0,f(x)\to 0$

Explanation of Solution

Given information:

A function $f$ is given.

  $f\left(x\right)=\frac{1-\cos x}{x}$

Notice that $f(x)$ is not defined when $x=0$ . What happens as $x$ approaches $0$ ?

Calculation:

Consider the function

  $f\left(x\right)=\frac{1-\cos x}{x}$

Now, sketch the graph of the function

  

Note that the given function has an $x$ in the denominator so $f(x)$ is not defined at $x=0$ .

However as $x$ approaches $0$ , so does $1-\cos x$ ,

Hence, as $x\to 0,f(x)\to 0$ .