Chapter 5.4, Problem 102E

a.

To determine

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

Answer to Problem 102E

  $\boxed{\text{all real numbers except }h=0}$

Explanation of Solution

Given information:

Let $x=\pi /3$ in the identity in Example 8 and define the functions $f$ and $g$ as follows.

  $f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}$

  $g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{1-\cosh }{h}\right)$

What are the domains of the functions $f$ and $g$ ?

Calculation:

Consider $x=\pi /3$ ,

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

From formula $f\left(h\right)=g\left(h\right)$

For domain $f\left(h\right)$ put denominator of $f\left(h\right)=0$ , $\therefore h=0$

So domain of $f\left(h\right)$ , $h$ =all real numbers except $h=0$

For domain $g\left(h\right)$ put denominator of $g\left(h\right)=0$ , $\therefore h=0$

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

So domain of $g\left(h\right)$ , $h$ =all real numbers except $h=0$

Hence, domain of $f$ and $g$

  $\boxed{\text{all real numbers except }h=0}$ .

b.

To determine

Use a graphing utility to complete the table.

Answer to Problem 102E

  

Explanation of Solution

Given information:

Let $x=\pi /3$ in the identity in Example 8 and define the functions $f$ and $g$ as follows.

  $f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}$

  $g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{1-\cosh }{h}\right)$

Use a graphing utility to complete the table,

  

Calculation:

The graph is as shown,

  

The table is as,

  

Hence the result is tabulated.

c.

To determine

Use the graphing utility to graph the functions $f$ and $g$ .

Answer to Problem 102E

  

  

Explanation of Solution

Given information:

Let $x=\pi /3$ in the identity in Example 8 and define the functions $f$ and $g$ as follows.

  $f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}$

  $g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{1-\cosh }{h}\right)$

Use the graphing utility to graph the functions $f$ and $g$ .

Calculation:

The graph of functions $f$ is as shown,

  

The graph of functions $g$ is as shown,

  

Hence the result is plotted.

d.

To determine

Use the table and the graphs to make a conjecture about the values of the functions $f$ and $g$

  $h\to 0^{+}$ .

Answer to Problem 102E

  $\boxed{\underset{h\to 0^{+}}{\lim }f\left(x\right)=0.00872,\underset{h\to 0^{+}}{\lim }g\left(x\right)=0.00872}$

Explanation of Solution

Given information:

Let $x=\pi /3$ in the identity in Example 8 and define the functions $f$ and $g$ as follows.

  $f\left(h\right)=\frac{\sin \left[\left(\pi /3\right)+h\right]-\sin \left(\pi /3\right)}{h}$

  $g\left(h\right)=\cos \frac{\pi }{3}\left(\frac{\sinh }{h}\right)-\sin \frac{\pi }{3}\left(\frac{1-\cosh }{h}\right)$

Use the table and the graphs to make a conjecture about the values of the functions $f$ and $g$

  $h\to 0^{+}$ .

Calculation:

From the table we observed that,

When $h\to 0^{+}$

  $f\left(h\right)$ and $g\left(h\right)$ near around $0.00872$

But when $h=0$ , $f\left(h\right)$ and $g\left(h\right)$ is undefined.

Hence, conjecture about the values of the functions $f$ and $g$ are,

  $\boxed{\underset{h\to 0^{+}}{\lim }f\left(x\right)=0.00872,\underset{h\to 0^{+}}{\lim }g\left(x\right)=0.00872}$