The number N of locations of a popular coffeehouse chain is...

The number $N$ of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.) $$\begin{array}{|c|c|c|c|c|c|} \hline \text { Year } & 2004 & 2006 & 2008 & 2010 & 2012 \\ \hline N & 8569 & 12,440 & 16,680 & 16,858 & 18,066 \\ \hline \end{array}$$ (a) Find the average rate of growth (i) from 2006 to 2008 (ii) from 2008 to 2010 In each case, include the units. What can you conclude? (b) Estimate the instantaneous rate of growth in 2010 by taking the average of two average rates of change. What are its units? (c) Estimate the instantaneous rate of growth in 2010 by measuring the slope of a tangent.

The number $N$ of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.)
$$\begin{array}{|c|c|c|c|c|c|}
\hline \text { Year } & 2004 & 2006 & 2008 & 2010 & 2012 \\
\hline N & 8569 & 12,440 & 16,680 & 16,858 & 18,066 \\
\hline
\end{array}$$
(a) Find the average rate of growth
(i) from 2006 to 2008
(ii) from 2008 to 2010
In each case, include the units. What can you conclude?
(b) Estimate the instantaneous rate of growth in 2010 by taking the average of two average rates of change. What are its units?
(c) Estimate the instantaneous rate of growth in 2010 by measuring the slope of a tangent.

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Key Concepts

Slope of a Tangent Line

The slope of a tangent line to a curve at a given point provides a geometric interpretation of the instantaneous rate of change at that point. This slope represents how steeply the function is rising or falling at that exact location and is equivalent to the derivative at that point.

Average Rate of Change

This concept refers to the change in a quantity over a given time interval divided by the duration of that interval. It provides a measure of how quickly the quantity increases or decreases on average during the period, and is often computed as the difference between two values divided by the difference in time.

Instantaneous Rate of Change

This concept represents the rate of change of a quantity at a specific moment in time, rather than over an interval. It is defined as the limit of the average rate of change as the time interval approaches zero, and is mathematically captured by the derivative.

Unit Analysis

In the context of rates of change, unit analysis involves ensuring that calculations correctly combine the units of the quantities involved, such as changes in the quantity (e.g., number of locations) per unit time (e.g., year). This check confirms that the resulting rate appropriately reflects the relationship between the measured variables.

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