Chapter 5.4, Problem 81E

(a)

To determine

The domains of the functions $f$ and $g$ .

Answer to Problem 81E

The domains of the functions $f$ and $g$ is all the real number for $h$ except $h=0$ .

Explanation of Solution

Given information:

The given functions $f$ and $g$ .

  $\begin{array}{l}f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}\\ g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{\left(1-\cosh \right)}{h}\right)\end{array}$

Calculation:

  $h$ is a denominator $h=0$ can’t be the domain because functions tends to infinity at $h=0$ .

For all other real numbers, $f$ and $g$ have values. The domain of $f$ and $g$ are all the real numbers except zero. The domain of function is all the real number for $h$ except $h=0$ .

Therefore, the domains of the functions $f$ and $g$ is all the real number for $h$ except $h=0$ .

(b)

To determine

Thevalues of the functions $f$ and $g$ at different value of $h$ to complete the table.

Answer to Problem 81E

The values of the functions $f$ and $g$ at different value of $h$ shown in table.

Explanation of Solution

Given information:

The table is shown below.

$h$	$0.01$	$0.02$	$0.05$	$0.1$	$0.2$	$0.5$
$f\left(h\right)$
$g\left(h\right)$

Calculation:

Graphing calculator to obtain the value of function $f\left(h\right)$ and $g\left(h\right)$ at different point as follows.

The value of the functions $f$ and $g$ at different value of $h$ .

$h$	$0.01$	$0.02$	$0.05$	$0.1$	$0.2$	$0.5$
$f\left(h\right)$	$0.4957$	$0.4913$	$0.4781$	$0.4559$	$0.4104$	$0.2674$
$g\left(h\right)$	$0.4957$	$0.4913$	$0.4781$	$0.4559$	$0.4104$	$0.2674$

Therefore,thevalues of the functions $f$ and $g$ at different value of $h$ shown in the above table.

(c)

To determine

The graph of the functions $f$ and $g$ .

Answer to Problem 81E

The graph of the functions $f$ and $g$ shown in the screen.

Explanation of Solution

Given information:

The given functions $f$ and $g$ .

  $\begin{array}{l}f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}\\ g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{\left(1-\cosh \right)}{h}\right)\end{array}$

Calculation:

Press $\left[\text{WINDOW}\right]$ and set the window size. Enter ${\text{x}}_{\text{min}}=\text{0,}{\text{X}}_{\text{max}}=\text{1,}{\text{X}}_{\text{scl}}=\text{0}\text{.1,}{\text{Y}}_{\text{min}}=\text{0,}{\text{Y}}_{\max }=1$ .

The screen look like as.

  

Press $\left[\text{GRAPH}\right]$ .

  

Both the functions have the same value the graph coincide to each other.

Therefore, the graph of the functions $f$ and $g$ shown in the screen.

(d)

To determine

The conjecture about the values of the functions $f$ and $g$ as $h\to 0^{+}$ .

Answer to Problem 81E

The conjecture about the values of the functions $f$ and $g$ as $h\to 0^{+}$ is $f\left(h\right)\to 0.5$ and $g\left(h\right)\to 0.5$ .

Explanation of Solution

Given information:

The given functions $f$ and $g$ .

  $\begin{array}{l}f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}\\ g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{\left(1-\cosh \right)}{h}\right)\end{array}$

Calculation:

Press $\left[\text{TRACE}\right]$ and then move the cursor to the left most point by pressing arrow keys, the graph as $h\to 0^{+}$ is as shown below.

  

From the graph and table as $h\to 0^{+},f\left(h\right)\to 0.5$ and $g\left(h\right)\to 0.5$ .

Therefore, the conjecture about the values of the functions $f$ and $g$ as $h\to 0^{+}$ is $f\left(h\right)\to 0.5$ and $g\left(h\right)\to 0.5$ .