## Functions

Functions are really pretty simple things since they express a
relationship between two distinct 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 your speed and the time 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. 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, there is a function which
relates the time after our action to the amount of pollution in the
river.
Calculus is vitally important for the sciences for it provides tools
to extract important information about functions.
Let's begin with a famous example. Pretend that you drop a ball and
that you would like to know how far it falls after a certain time. A
convenient way to understand this would be to use a strobe light to
take a picture of where the ball is at regular intervals after being
released.
******LEAH'S FALLING BALL APPLET HERE*******
If we only check every ten seconds, we don't have a very good feel for
where the ball is after one second. So we could check every second to
see where the ball is. Still, we don't have a good feel for where the
ball would be after one one-tenth of a second. Again, we could
increase the frequency with which we check. You can imagine how this
goes: to get a better understanding of where the ball is, you can
simply make more measurements. But no matter how many measurements
you make, you will not know what happens at * every * time.
This is where something very beautiful happens. If we put the data
points on a graph like in the picture above, we can see that the
points form a pattern. They all seem to lie on a curve and the
property of this curve is that the vertical distance fallen, measured
in meters, seems to always be given by 4.9 times the square of the
time since release, measured in seconds. At this point, this is just
an observation which we have not explained (we will do that later).
However, once we make this observation, we can make * predictions
* about where we think the ball should be. For instance, after
0.001 seconds, we expect that the ball would only have fallen
0.0000049 meters. This is something we could verify if our strobe
light allowed us.
From this, we are lead to write something like y = 4.9t^2 to denote
the relationship between t, the time since the ball was released, and
y, the vertical distance the ball has fallen.
This example illustrates something important about functions.
We can try to understand functions that describe scientific processes
by making some measurements. In general, this cannot tell us the
whole story but rather only what is going on when we happen to be
looking (that is, making a measurement). However, if we make enough
measurements, we often develop confidence about what is happening
between the measurements so that we feel like we understand the entire
process.
The figure below graphs the deceleration of the Mars Pathfinder versus
time. Of course, the Pathfinder only reported it deceleration to the
scientists back on Earth every so often. This graph was produced by
just connecting the dots between the data points. Still, we feel
fairly confident that the graph reflects, with some accuracy, the
deceleration between the data points. Notice that, for this function,
it would be difficult to write down a nice algebraic expression which
expresses this relationship. Nevertheless, this is a perfectly good
function.
(The large bump in the graph occurred when Pathfinder entered the
Martian atmosphere. The second, smaller bump occurred when the
parachutes were deployed.)
In this class, we will encounter function in both of these forms. At
some times, we will assume that the data about our function is
obtained from just making some measurement. At others, we will assume
that we can understand what happens at every time and write things
like y=4.9t^2.