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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?

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In economics, a firm wants to maximize profit. Your ice cream shop has some costs that vary depending on how many half-gallon ice cream packages are made, and some benefits, the revenue from selling ice cream. The additional cost of making one more package of ice cream is called the marginal cost, and it can change based on how much ice cream you are already making. The marginal benefit of course the price that you sell each package for.

In order to maximize profit, you have to choose some quantity of ice cream packages to produce. You do this by setting marginal cost equal to marginal benefit, and then solving for production. So if you have a formula for what the price of ice cream will be over time, then you can predict your marginal benefit of production over time, and can anticipate what your production will look like in the future.

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The words may not be as profound as some other economist would say, but predicting revenues is part of econometrics and that is important for a wide range of reasons. Also, in your example, keep in mind that inflation takes a very special role in Economics. It is not explicit what $r^2$ is but it seems that example, for instance, does not consider inflation rates.

But rates of change can be considered in strategic decision making (microeconomics), to know the inflation rate (macroeconomics) and many other dimensions that I cannot even think of.

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