Two Faces of Functions
UBC Calculus Online Course Notes

Functions: Stars of the Show

The starring roles of this course are played by functions. Functions are pretty simple things---they just express a relationship between two different quantities. Perhaps you have thought, "the faster I drive, the quicker I'll arrive at UBC." If so, then you understand the idea of a function. You are stating that there is a relationship between the speed at which you drive and the time it takes before you arrive at your destination.

It's difficult to overstate the importance of functions in science since science aims to understand the relationship between different quantities and to make predictions based on this understanding. For instance, if we have a polluted river and take some steps to clean it up, we will need to judge how effective our methods are. To do this, we might like to know how polluted the river is so many days after we take action. In this situation, there is a function which relates time to the amount of pollution in the river. We might like to understand this relationship so that we could predict when the river will no longer be dangerous for the fish in it.


Notation

Let's continue with this example just to introduce some terminology. If we let t measure the amount of time and p measure the amount of pollution in the river, we will write p(t) to refer to the function which relates these two quantities. You should think about this as saying that p depends on t and if we choose a time, say 10 days, then the function will give us the value p(10) (don't worry about how it does this yet) which is the amount of pollution after 10 days.


The main reason that Calculus is so important for the sciences, and the reason you are probably taking this course, is that it provides powerful tools to extract important information about functions. In this unit, we will explore some of the ways in which functions arise.


For your consideration:

  • Can you recognize something you heard about on the radio or in the newspaper in the last few days as a function?

  • What is the relationship being represented?

  • What kind of prediction would be important to make based on an understanding of this function?