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