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22 September 2001

The Study of Change

by George E. Hrabovsky, President of MAST

News from MAST

Hello everyone. Another week has gone by and the entire world has changed from what it was. We are talking about war and terror. We are all choked up watching TV and hanging flags in honor of the dead. Allow me to start by offering my sympathies to the families and friends of those who have been lost; I know what they are going through.   To those who did this, and those who are involved in any way, I have only one thing to say, "We will get you for this."

Today, since we are seeing such sweeping changes, I will discuss how we study changes. I recommend that you grab some paper and a pen/pencil and follow along step-by-step.

An Introduction to Rates of Change

Unless the situation you are investigating is trivially boring you will determine that a function will change with respect to a variable. Let us say that we are examining a function of time. The data we collect indicates that the dependence of the variable, say y, has the form,

If we are to understand this relationship, we need to understand how y changes with respect to t. The rate of change of a function is equal to the slope of the function. For a linear function this is easy to find, but for a nonlinear function, like that above, it requires some special techniques.

Recall that the slope of a line is, "Rise over run." What this means is that is we increase t by a small quantity, traditionally , and then we determine

We can also write y as f(t). The "run" is also called the interval of t and is determined by,

The "rise" is the written as,

Thus, "rise over run" is,

This is called a divided difference.

Now we can look at . What is it really? What is ? If we think about it we see that we can write as,

This being true, we can rewrite as or, in our case,

By the Binomial Theorem we can rewrite this,

In order to find f(t) we need to subtract our last result by f(t), or,

Since the is subtracting itself it cancels out, so we are left with,

To complete the divided difference (the slope) we need to divide this by t. We do this and get,

Carrying out this division, we have,

This, then, is the slope of the function. Its rate of change. Unfortunately we have a conundrum; what is t?

The answer has at its root the idea of limits that I introduced last week. If we narrow the limit of a variable to a smaller and smaller value, the function will get closer and closer to its true value. What we really need to look for is an infinitely small change in value from one value of t to the next. Another way of saying that is that we need to see what happens in the limit where t vanishes (the distance between points is zero). So, we write,

Now it reasonable to ask the question, "What do I do now?" You replace every occurence of t by its limit, in this case 0. Thus we get,

So, the rate of change of our function in time is 2t.

What we have just done is the calculate a derivative. This is a full third of calculus. We can write this in a shorter way, . So, we write,

In fact, the general definition of a derivative is,

 Try calculating some of your own.

 

Books I Like
Richard Courant, Fritz John, 1989, Introduction to Calculus and Analysis, Volume 1, Springer. This is a very nice calculus book, but the math is pretty heavy. I recommend this for the serious theorist who wants to really learn what makes calculus work.

Morris Kline 1977, Calculus An Intuitive and Physical Approach, John Wiley and Sons (Dover Publications reprint 1998). This is a pretty good book and it has the advantage of being very inexpensive.

Alexander J. Hahn 1998, Basic Calculus, Springer-Verlag, New York. This is a wonderful treatment of calculus from its early roots in Greek astronomy, through Newton and Leibnitz, to some very iteresting applications. A very unconventional approach, but it gets the job done in a very interesting way.

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Converted by Mathematica      September 19, 2001