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I'm Big Bryce, Li'l Bryce's grampa. Missy tells me you are preparing for a calculus exam, and she thought that I may be able to help a little.

I don't know how far along you are, but there is a marvelous little book titled Calculus Made Easy by Slyvanus P. Thompson. Almost without exception, and this book is the exception that proves the rule, mathematics teachers especially will go off their collective bonkers if they find you are using a made easy book, of any kind for that matter.

Thompson, however, wrote his book for budding engineers and believed that the rigorous approach of typical mathematics texts was not what they needed. He believed they should dig right in and start using calculus right away. Indeed, he writes in an apolog. "Now, if the professional mathematicians aren't too lazy, they will rise up as one man and damn this as a thoroughly bad book. They will say the reason that it is easy is that the author left out all the parts of calculus; us that are truly difficult. And the ghastly fact about this accusation is that it is true. Any why not. You don't deny a child the use of a language just because he doesn't know all the rules of syntax. It would be equally ridiculous to deny a young engineer the use of these powerful means of reckoning simply because he doesn't know the irrelevant mathematical gymnastics that the professional mathematicians are so proud of."

Needless to say, his comments didn't make him too popular with the British Mathematical Society, and they banned him.

However, the printing of the book that I first had was its 26th, and I first obtained the book back in the early 1950s. The book went out of print for awhile, and someone made off with my copy. I think it was my brother when he was in college. The book was reprinted again in the late 1960s and I was able to get another copy which I've lent to many young engineers over the years. I believe it is still available through Barns and Nobel ( or ( And you might just be able to find it in your local library, and if not have your library order it through their lending services. They can get it for you one way or another.

The book has as its initial dedication, "What one fool can do another can." It then presents an epilog, which starts out "To Deliver you from the Preliminary Terrors." Thompson talks about dx which he says simply means a little bit of x. Then he shows an integral sign which he says is simply a long S and means to sum up. The professional mathematicians prefer to say take the integral of. Obviously, he says, that if you sum up all the little bits of x. that is, Sdx you will get x. And you have just performed your first integration.

Now in a regular textbook you will go through development of the concept of limits of approximations to the number you are trying to obtain. The number of course won't be explicit, such as 3,4, or any other number 3.1415926535...., but will be described as a generality, that is, as an equation or an algebraic expression. For example the y in, y = mx + b is really a number that you get by applying the rule mx + b. This is the equation for a straight line of course, but it has particular relevance in Calculus.

If I integrate the expression dy = mx + b I'll obtain y = m/2 x^2 + bx + C which has the form a x squared plus bx plus a constant. You recognize this as the quadratic equation which can be use to describe a parabolic curve. The dy = mx + b looks like y = mx + b, our straight line. And, indeed it is, but it is only a tiny bit of a straight line.

If you start with an equation for a parabolic curve and you take its derivative you get an expression for a straight line. The equation we get for the straight line has particular relevance. It tells us how fast the parabolic curve is changing at any given place along its path. We can get an explicit rate of change value simply by substituting in x and y numbers (values) taken from a particular point of interest anywhere along the parabolic curve. Engineers and Physicist do this sort of thing all the time.

They perform experiments and plot the data. Then they use other common mathematical methods which you will learn later to obtain equations. They then use the methods of calculus to analyze the data. They want to know such things as how fast some object is turning or twisting so they can calculate the forces and stresses being put on the object.

It may not be possible to directly observe many of these turns and twists, but it is pretty easy to calculate them if you have the basic plotted data. In this way, engineers and physicists can build and design stuff that won't break. We could, of course, simply build it much stronger and heavier than necessary and be pretty sure it wouldn't break. But, suppose we were trying to build a rocket ship. If we built it too heavy, we wouldn't be able to get it off the ground. So that's where your mathematics comes in. It allows you to build something optimally. Not too strong, but not too weak.

Economists, also use calculus extensively. They want to know how the "forces" of buying, trading, selling stocks or the rate of accumulation of surpluses, or the expenditure of money, or rates of taxation, and on and on affect the economy.

Heck, even the psychologist use calculus. They are constantly trying to analyze surveys and other sample data that they acquire. They have to determine how quickly it changes at any given time, and they have to summarize (integrate) it into some manageable form.

You are undertaking an exciting adventure into how the world really works. Calculus is the tool that opens the window to allow you to observe what is normally unobservable by sight alone. There are two major transitions in learning mathematics. The first is when you make the association that letters can stand for numbers, algebra, and the second is when you move from the static world of mathematics, algebra, plane geometry, etc., to the dynamic world of Calculus. By dynamic we mean that calculus is the mathematics of things in motion.

Things in motion doesn't necessarily mean physical objects. We also think of the abstraction of imagining a point to be moving along a line (curved or straight) as a thing in motion. And the neat thing is that both can be exactly describe by calculus, as can many other kinds of happenings.

I've rambled on for a long time now, Grandpas do that. But do let me know how you are coming along. There is a wealth of tutorial material available on the Internet, and it's a lot of fun tracking it down. You should be able to find all kinds of Calculus tutorials from practically every college and university. Instructors, today, often put their lessons and exams on the net and publish the solutions to homework problems.

There is no substitute, however, for finding someone to study with. And it doesn't matter that it is the blind leading the blind. What does matter is having someone to bounce off your ideas about and understanding of the material you're studying.

If you can't find anyone, I will be glad to provide any assistance I can.

Have a good day and good calculating.

Grandpa Bryce

© 1999 Round Persons' Enterprises


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