Chapter 2.6, Problem 44E

i.

To determine

The average rate of coffee house growth from $2005\text{ to 2007}$ .

Explanation of Solution

Given information:

The number N of locations of a coffee house chain is shown in table.

$\text{Year}$	$2003$	$2004$	$2005$	$2006$	$2007$
$N$	$7225$	$8569$	$10241$	$12440$	$15011$

Calculations:

Here the growth of coffee house from $2005\text{ to 2006}$ is $12440-10241=2199$

Here the growth of coffee house from $2006\text{ to 2007}$ is $15011-12440=2571$

Here the average growth of coffee house from $2005\text{ to 2007}$ is $\frac{2199+2571}{2}=2385$

ii.

To determine

The average rate of coffee house growth from $2005\text{ to 2006}$

Explanation of Solution

Given information:

The number N of locations of a coffee house chain is shown in table.

$\text{Year}$	$2003$	$2004$	$2005$	$2006$	$2007$
$N$	$7225$	$8569$	$10241$	$12440$	$15011$

Calculations:

Number of coffee house in $2005$ is $10241$ .

Number of coffee house in $2006$ is $12440$ .

Here the growth of coffee house from $2005\text{ to 2006}$ is $12440-10241=2199$

iii.

To determine

The average rate of coffee house growth from $2004\text{ to 2005}$

Explanation of Solution

Given information:

The number N of locations of a coffee house chain is shown in table.

$\text{Year}$	$2003$	$2004$	$2005$	$2006$	$2007$
$N$	$7225$	$8569$	$10241$	$12440$	$15011$

Calculations:

The number of cell phone subscribers in $2004$ is $8569$ .

The number of cell phone subscribers in $2005$ is $10241$ .

Thus the growth of cell phone users from $2004\text{ to 2005}$ is $10241-8569=1672$

iv.

To determine

The instantaneous rate of growth in $2005$ by taking the average of two average rate of change.

Explanation of Solution

The number N of locations of a coffee house chain is shown in table.

$\text{Year}$	$2003$	$2004$	$2005$	$2006$	$2007$
$N$	$7225$	$8569$	$10241$	$12440$	$15011$

Calculations:

Thus the average number of coffee house from $2004\text{ to 2005}$ is $\frac{10241+8569}{2}=9405$

Therefore the instantaneous rate of growth in $2005$ is $10241-9405=836$

v.

To determine

The instantaneous rate of growth in $2005$ by measuring the slope of tangent.

Explanation of Solution

The number N of locations of a coffee house chain is shown in table.

$\text{Year}$	$2003$	$2004$	$2005$	$2006$	$2007$
$N$	$7225$	$8569$	$10241$	$12440$	$15011$

Calculations:

The slope of the tangent is given the formula,

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

  $\text{Slope=}\frac{10241-8569}{1}=1672$