Relating these representations of functions
UBC Calculus Online Course Notes

Important information contained in functions

As we've said earlier, Calculus is very helpful for understanding functions. However, before we jump into that, let's illustrate two kinds of information we would like to get out of functions.

Quantitative information

Functions can help us make decisions by giving numerical answers to questions such as "When?" and "How much?" For example, we will write C(t) to refer to the function which tells you the cost C of a pound of coffee at a particular time t. Now suppose one morning you wake up, discover you are out of coffee and rush out to buy some. So that you take enough money to the store, it is important to know the value of C at that particular instant. This is an example of the kind of quantitative information functions provide.

Qualitative information

Just as important though is qualitative information provided by the function. This is information which doesn't depend on precise numbers but rather a more general understanding of the function.

Let's continue thinking about our function C(t) which describes the cost C of a pound of coffee at some time t. Suppose tomorrow morning you wake up and realize you are getting low on coffee. You have enough for today, but you will definitely need to buy more in the next few days. If you have a general understanding of the function C(t), you will notice that the price is going up; that is, as t increases, the value of C increases too. This means that if you wait, you'll spend more money than you really have to and you won't have any money for doughnuts. Realizing this, you rush right out and buy coffee today. (See, isn't Calculus useful?)

In this situation, it's not so important what the cost C of coffee is right now, but rather the fact that C is increasing with time.

In the future, we will see how Calculus gives us the ability to deduce both quantitative and qualitative information.