Chapter 2.6, Problem 43E

i.

To determine

The average rate of cell phone growth from $2002\text{ to 2006}$ .

Explanation of Solution

Given information:

The number N of US cell phone subscribers (in millions) is shown in table.

$t$	$1996$	$1998$	$2000$	$2002$	$2004$	$2006$
$N$	$44$	$69$	$109$	$141$	$182$	$233$

Calculations:

Here the growth of cell phone users from $2002\text{ to 2004}$ is $182-141=41$

Thus the average growth of cell phone users from $2002\text{ to 2004}$ is $\frac{41}{2}=20.5$

Here the growth of cell phone users from $2004\text{ to 2006}$ is $233-182=51$

Here the average growth of cell phone users from $2004\text{ to 2006}$ is $\frac{51}{2}=25.5$

Therefore the average rate of cell phone growth from $2002\text{ to 2006}$ is $\frac{20.5+25.5}{2}=\frac{46}{2}=23$

ii.

To determine

The average rate of cell phone growth from $2002\text{ to 2004}$

Explanation of Solution

Given information:

The number N of US cell phone subscribers (in millions) is shown in table.

$t$	$1996$	$1998$	$2000$	$2002$	$2004$	$2006$
$N$	$44$	$69$	$109$	$141$	$182$	$233$

Calculations:

The number of cell phone subscribers in $2002$ is $141$ .

The number of cell phone subscribers in $2004$ is $182$ .

Thus the growth of cell phone users from $2002\text{ to 2004}$ is $182-141=41$

Thus the average growth of cell phone users from $2002\text{ to 2004}$ is $\frac{41}{2}=20.5$

iii.

To determine

The average rate of cell phone growth from $2000\text{ to 2002}$

Explanation of Solution

Given information:

The number N of US cell phone subscribers (in millions) is shown in table.

$t$	$1996$	$1998$	$2000$	$2002$	$2004$	$2006$
$N$	$44$	$69$	$109$	$141$	$182$	$233$

Calculations:

The number of cell phone subscribers in $2000$ is $109$ .

The number of cell phone subscribers in $2002$ is $141$ .

Thus the growth of cell phone users from $2000\text{ to 2002}$ is $141-109=32$

Thus the average growth of cell phone users from $2000\text{ to 2002}$ is $\frac{32}{2}=16$

iv.

To determine

The instantaneous rate of growth in $2002$ by taking the average of two average rate of change.

Explanation of Solution

Given information:

The number N of US cell phone subscribers (in millions) is shown in table.

$t$	$1996$	$1998$	$2000$	$2002$	$2004$	$2006$
$N$	$44$	$69$	$109$	$141$	$182$	$233$

Calculations:

Thus the average growth of cell phone users from $2000\text{ to 2002}$ is $\frac{32}{2}=16$

Therefore the instantaneous rate of growth in $2002$ is $141-(109+16)=16$

v.

To determine

The instantaneous rate of growth in $2002$ by measuring the slope of tangent.

Explanation of Solution

Given information:

The number N of US cell phone subscribers (in millions) is shown in table.

$t$	$1996$	$1998$	$2000$	$2002$	$2004$	$2006$
$N$	$44$	$69$	$109$	$141$	$182$	$233$

Calculations:

Thus the average growth of cell phone users from $2000\text{ to 2002}$ is $\frac{32}{2}=16$

The slope of the tangent is given the formula,

  $f\text{'}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

  $\text{Slope=}\frac{\text{141}-(109+16)}{1}=16$