Relating these representations of functions

It is important to understand the relationship between both ways of discussing functions.

An Example

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. Click in the box below and you will see what happens when we flash the strobe every five seconds. As you can see, this gives us a collection of data points which tell us where the ball is every five seconds. However, we might not feel very confident about where the ball is after one second (maybe it visits Moose Jaw in between our measurements). So we could do the experiment again and check where the ball is every second. 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 interesting happens. If we put the data points on a graph as in this picture, we can see that the points form a pattern. They all seem to lie on a parabola with the property 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 stage, 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 might expect that the ball would only have fallen 0.0000049 meters. This is something we could verify with our strobe light. From this, we are lead to write something like y = 4.9t2 to denote the relationship between t, the time since the ball was released, and y, the vertical distance the ball has fallen.

For your consideration:

• What kinds of measurements can we make to figure out if the trajectory of the ball is really smooth or just a lot of jumps at very short intervals?
• What is our visual system doing to make a film or video clip (which is really discrete) seem to be smooth?
• Where do you think the ball would be after one year?

When answering this last question, you need to think about the process that the function is describing. What are the assumptions you made to arrive at the answer that you gave? This is an important part of scientific thinking---when making a prediction, you must always consider whether there will be changes in the process which invalidate your assumptions. I would imagine that, in one year, the ball will be sitting neglected in a closet somewhere (sniff!).

Conclusions

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 (or maybe after the measurements end) so that we feel like we understand the entire process and can hence make predictions. In this class, we will encounter functions in both of these forms. Sometimes, we will assume that the data about our function is obtained from just making some measurements. Other times, we will assume that we can understand what happens at every time and write things like y=4.9t2. Eventually, we will see why it is useful to be able to think in both ways.