Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 42

The table below represents the number $N$ of locations of a popular coffee house.

$
\begin{array}{|c|c|c|c|c|c|}
\hline\\
Year & 1998 & 1999 & 2000 & 2001 & 2002\\
\hline
N & 1886 & 2135 & 3501 & 4709 & 5886\\
\hline
\end{array}
$


a.) Determine the average rate of growth or the following fire frames.


$
\begin{equation}
\begin{aligned}
& (i) 1999 \text{ to } 2000 \quad (iii) 2000 \text{ to } 2002\\
& (ii) 2000 \text{ to } 2001
\end{aligned}
\end{equation}
$


b.) Estimate the instantaneous rate of growth in 2000 by taking the average of two average rates of change.

c.) Estimate the instantaneous rate of growth in 2000 by measuring the slope of a tangent.

a.) $(i)$ from 1999 to 2000.

$\quad \displaystyle \text{average rate } = \frac{N(2000) - N(1999)}{2000-1999} =
\frac{3501 - 2135}{2000 - 1999} = 1366 \frac{\text{number of locations}}{\text{year}}$

$(ii)$ from 2000 to 2001

$\quad \displaystyle \text{average rate } = \frac{N(2001) - N(2000)}{2001-2000} =
\frac{4709 - 3501}{2001 - 2000} = 1208 \frac{\text{number of locations}}{\text{year}}$

$(iii)$ from 2000 to 2002

$\quad \displaystyle \text{average rate } = \frac{N(2002) - N(2000)}{2002-2000} =
\frac{5886 - 3501}{2002 - 2000} = 1192.5 \frac{\text{number of locations}}{\text{year}}$


b.) We take the values of $(i)$ and $(ii)$ from part(a) for the instantaneous rate as
$\displaystyle \frac{1366+1208}{2} = 1287 \frac{\text{number of locations}}{\text{year}}$


c.)






Referring to the graph, instantaneous rate of growth in year 2000 can be estimated as 1237.5
$\displaystyle \frac{\text{number of locations}}{\text{year}}$