I'd like to introduce math-students to the concept of average and instantaneous rate of change, in order to get them interested in the topic. I found the following example:
Based on data from 1990 to 2000, the average price for a half gallon of prepackaged ice cream may be modeled by $$ C(t) = 0.0142*t^2 - 0.0272*t + 2.53 $$ where t is the number of years since 1990. (Source: Modeled from Statistical Abstract of the United States, 2001, Table 706, p. 468.) To help project future revenue, a creamery wants to know what the average annual increase in the price of ice cream was from 1990 to 2000 and at what rate the price will be increasing at the end of 2000.
So, in other words, we are looking for the average and instantaneous rate of change.
My question is the following:
Why is the rate of change important for economics? I know this sounds silly, but if my goal was to project the price of ice cream for let's say 2015, all I had to do was insert the value 25 for t and I'd be done, right? So why do we care about the rate of change in economics? What is to be gained by this information and what would be decided differently after finding out the rate of change?