Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 41

The table below represents the estimated percentage $P$ of the population of cell phone users in Europe.



$
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Year} & 1998 & 1999 & 2000 & 2001 & 2002 & 2003\\
\hline\\
P & 28 & 39 & 55 & 68 & 77 & 83\\
\hline
\end{array}
$



a.) Determine the average rate of cell phone growth on the following time frames



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



$\displaystyle \text{average rate} = \frac{P(2000)-P(1999)}{2000-1999} = \frac{55-39}{2000-1999} = 16 \frac{\%}{\text{year}}$



$(ii)$ from 2000 to 2001



$\displaystyle \text{average rate} = \frac{P(2001)-P(2000)}{2001-2000} = \frac{68-55}{2001-2000} = 13 \frac{\%}{\text{year}}$



$(iii)$ from 2000 to 2002



$\displaystyle \text{average rate} = \frac{P(2002)-P(2000)}{2002-2000} = \frac{77-55}{2002-2000} = 11 \frac{\%}{\text{year}}$



b.) The value of instantaneous rate of growth in 2000 can be computed as $\displaystyle \frac{16+13}{2} = 14.5 \frac{\%}{\rm{year}}$



c.)






Referring to the graph, instantaneous rate of growth in year 2000 can be estimated as $14.3 \%$ per year.