Silvanus P. Thompson

What Is a Function

When we asked the question What is a number? we certainly did not expect an answer such as 1 and π are numbers.

Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus

Prologue To Deliver you from the Preliminary Terrors On Different Degrees of Smallness On Relative Growings Simplest Cases Next Stage: What to do with Constants Sums, Differences, Products and Quotients Successive Differentiation When Time Varies Introducing a Useful Dodge Geometrical Meaning of Differentiation Maxima and Minima Curvature of Curves Other Useful Dodges On True Compound Interest and the Law of Organic Growth How to Deal with Sines and Cosines Partial Differentiation Integration Integrating as the Reverse of Differentiating On Finding Areas by Integration Dodges, Pitfalls and Triumphs Finding Solutions A Little More about Curvature of Curves How to Find the Length of an Arc on a Curve Epilogue and Apologue

Partial Fractions and Inverse Functions

We have seen that when we differentiate a fraction we have to perform a rather complicated operation; and, if the fraction is not itself a simple one, the result is bound to be a complicated expression. If we could split the fraction into two or more simpler fractions such that their sum is equivalent to the original fraction, we could then proceed by differentiating each of these simpler expressions. And the result of differentiating would be the sum of two (or more) derivatives, each one of which is relatively simple; while the final expression, though of course it will be the same as that which could be obtained without resorting to this dodge, is thus obtained with much less effort and appears in a simplified form.

Introducing a Useful Dodge

Sometimes one is stumped by finding that the expression to be differentiated is too complicated to tackle directly.

When Time Varies

Some of the most important problems of the calculus are those where time is the independent variable, and we have to think about the values of some other quantity that varies when the time varies. Some things grow larger as time goes on; some other things grow smaller. The distance that a train has travelled from its starting place goes on ever increasing as time goes on. Trees grow taller as the years go by. Which is growing at the greater rate: a plant 12 inches high which in one month becomes 14 inches high, or a tree 12 feet high which in a year becomes 14 feet high?

On Finding Areas by Integrating

One use of the integral calculus is to enable us to ascertain the values of areas bounded by curves.

Epilogue and Apologue

It may be confidently assumed that when this book Calculus Made Easy falls into the hands of the professional mathematicians, they will (if not too lazy) rise up as one man, and damn it as being a thoroughly bad book. Of that there can be, from their point of view, no possible manner of doubt whatever. It commits several most grievous and deplorable errors.

What Is a Limit

It is possible, though difficult, to understand calculus without a firm grasp on the meaning of a limit. A derivative, the fundamental concept of differential calculus, is a limit. An integral, the fundamental concept of integral calculus, is a limit.

What Is a Derivative

In Chapter 3 Thompson makes crystal clear what a derivative is, and how to calculate it. However, it seemed to me useful to make a few introductory remarks about derivatives that may make Thompson’s chapter even easier to understand.

On Relative Growings

All through the calculus we are dealing with quantities that are growing, and with rates of growth. We classify all quantities into two classes: constants and variables. Those which we regard as of fixed value, and call constants, we generally denote algebraically by letters from the beginning of the alphabet, such as a, b, or c; while those which we consider as capable of growing, or (as mathematicians say) of “varying”, we denote by letters from the end of the alphabet, such as x, y, z, u, v, w, or sometimes t.