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See "The Study of Change" (Mind of a Theorist, 21 September) and "Adding It All Together" (28 September)

05 October 2001

Summing It All Up

by George E. Hrabovsky, President of MAST

News from MAST

I was very alarmed by Shawn's email from yesterday. I want the membership to know that we intend to soldier on, even if SAS shuts down.

This week I intend to complete the whirlwind tour of calculus that started three weeks ago.

How to Come Close to Evaluating a Function

Like the columns of the last two weeks, it is best if you follow along with a pen and paper. It might be helpful to review the last three columns.

Have you ever wondered how a calculator comes up with a sine function or some other elementary special function? The answer is that it approximates the function. This column will show you how to do the same.

We begin by stating that the function we want to approximate is,

How do we go about approximating this? If we have any function of a variable, say t, we can approximate its behavior within a small radius from t traditionally called a. So we now have . It turns out that this approximation is given by something called a Taylor Series. Here is the formula for a general Taylor Series,

In order to understand this formula we need to understand what the symbols mean.

Let us say that we will choose n to be 3. This means we have a third-order series. For our example we have,

We now have another choice to make, what is a? If we look at it geometrically, where a is the radius of the neighborhood around t then we know that the smaller a is the closer to we will get. It seems reasonable to choose a as 0, this gives us,

Try this with a few other functions.

 

Books That I Like

Richard Courant, Fritz John, 1989, Introduction to Calculus and Analysis, Volume 1, Springer. This covers the concepts of Taylor series in specific and infinite series in general. It is very detailed and is a very good treatment of the subject.

Morris Kline 1977, Calculus An Intuitive and Physical Approach, John Wiley and Sons (Dover Publications reprint 1998). Has a decent chapter on the subject of series.

Murray R. Speigel, 1968, Schaum's Outline Mathematical Handbook, McGraw-Hill Inc. This book is now in its 34th printing, and is immensely useful. I highly recommend it, if you do not have access to Mathematica (and even if you do, it is a good resource). It is also imexpensive (less than $15). It has tables of derivatives and of various series.

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