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Title: Calculus Made Easy Being a very-simplest introduction to those beautiful methods which are generally called by the terrifying names of the Differentia
Author: Silvanus Thompson
Release Date: October 9, 2012 [eBook #33283] Most recently updated: November 18, 2021
Language: English
Character set encoding: UTF-
*** START OF THIS PROJECT GUTENBERG EBOOK CALCULUS MADE EASY ***
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MACMILLAN AND CO., Limited LONDON : BOMBAY : CALCUTTA MELBOURNE
THE MACMILLAN COMPANY NEW YORK : BOSTON : CHICAGO DALLAS : SAN FRANCISCO
THE MACMILLAN CO. OF CANADA, Ltd. TORONTO
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.
BY
F. R. S.
SECOND EDITION, ENLARGED
MACMILLAN AND CO., LIMITED
ST. MARTIN’S STREET, LONDON
What one fool can do, another can. (Ancient Simian Proverb.)
PREFACE TO THE SECOND EDITION.
The surprising success of this work has led the author to add a con- siderable number of worked examples and exercises. Advantage has also been taken to enlarge certain parts where experience showed that further explanations would be useful. The author acknowledges with gratitude many valuable suggestions and letters received from teachers, students, and—critics.
October, 1914.
CALCULUS MADE EASY viii
- XV. How to deal with Sines and Cosines Chapter Page - XVI. Partial Differentiation
- XVII. Integration
- XVIII. Integrating as the Reverse of Differentiating - XIX. On Finding Areas by Integrating - XX. Dodges, Pitfalls, and Triumphs - XXI. Finding some Solutions - Table of Standard Forms - Answers to Exercises.
PROLOGUE.
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks. Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics—and they are mostly clever fools—seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way. Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
CALCULUS MADE EASY 2
same thing as the whole of x). The word “integral” simply means “the whole.” If you think of the duration of time for one hour, you may (if you like) think of it as cut up into 3600 little bits called seconds. The whole of the 3600 little bits added up together make one hour. When you see an expression that begins with this terrifying sym- bol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow. That’s all.
CHAPTER II.
ON DIFFERENT DEGREES OF SMALLNESS.
We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness. We shall have also to learn under what circumstances we may con- sider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness. Before we fix any rules let us think of some familiar cases. There are 60 minutes in the hour, 24 hours in the day, 7 days in the week. There are therefore 1440 minutes in the day and 10080 minutes in the week. Obviously 1 minute is a very small quantity of time compared with a whole week. Indeed, our forefathers considered it small as com- pared with an hour, and called it “one minute,” meaning a minute fraction—namely one sixtieth—of an hour. When they came to re- quire still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth’s days, they called “second minutes” (i.e. small quantities of the second order of minute- ness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called. Now if one minute is so small as compared with a whole day, how
DIFFERENT DEGREES OF SMALLNESS 5
regard (^1) , 0001 , 000 (or one millionth) as a small quantity, then (^1) , 0001 , 000 of
1 , 0001 , 000 , that is^1 , 000 , 0001 , 000 , 000 (or one billionth) will be a small quantity of the second order of smallness, and may be utterly disregarded, by comparison. Then we see that the smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders, if only we take the small quantity of the first order small enough in itself. But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred. Now in the calculus we write dx for a little bit of x. These things such as dx, and du, and dy, are called “differentials,” the differential of x, or of u, or of y, as the case may be. [You read them as dee-eks, or dee-you, or dee-wy.] If dx be a small bit of x, and relatively small of itself, it does not follow that such quantities as x · dx, or x^2 dx, or ax^ dx are negligible. But dx × dx would be negligible, being a small quantity of the second order. A very simple example will serve as illustration. Let us think of x as a quantity that can grow by a small amount so as to become x + dx, where dx is the small increment added by growth. The square of this is x^2 + 2x · dx + (dx)^2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x^2. Thus if we
CALCULUS MADE EASY 6
took dx to mean numerically, say, 601 of x, then the second term would be 602 of x^2 , whereas the third term would be 36001 of x^2. This last term is clearly less important than the second. But if we go further and take dx to mean only 10001 of x, then the second term will be 10002 of x^2 , while the third term will be only (^1) , 0001 , 000 of x^2.
x
x
Fig. 1. Geometrically this may be depicted as follows: Draw a square (Fig. 1) the side of which we will take to represent x. Now suppose the square to grow by having a bit dx added to its size each way. The enlarged square is made up of the original square x^2 , the two rectangles at the top and on the right, each of which is of area x · dx (or together 2x · dx), and the little square at the top right-hand corner which is (dx)^2. In Fig. 2 we have taken dx as quite a big fraction of x—about 15. But suppose we had taken it only 1001 —about the thickness of an inked line drawn with a fine pen. Then the little corner square will have an area of only (^10) ,^1000 of x^2 , and be practically invisible. Clearly (dx)^2 is negligible if only we consider the increment dx to be itself small enough. Let us consider a simile.
CALCULUS MADE EASY 8
“So, Nat’ralists observe, a Flea “Hath smaller Fleas that on him prey. “And these have smaller Fleas to bite ’em, “And so proceed ad infinitum.” An ox might worry about a flea of ordinary size—a small creature of the first order of smallness. But he would probably not trouble himself about a flea’s flea; being of the second order of smallness, it would be negligible. Even a gross of fleas’ fleas would not be of much account to the ox.
CHAPTER III.
ON RELATIVE GROWINGS.
All through the calculus we are dealing with quantities that are grow- ing, 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 be- ginning 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 de- note by letters from the end of the alphabet, such as x, y, z, u, v, w, or sometimes t. Moreover, we are usually dealing with more than one variable at once, and thinking of the way in which one variable depends on the other: for instance, we think of the way in which the height reached by a projectile depends on the time of attaining that height. Or we are asked to consider a rectangle of given area, and to enquire how any increase in the length of it will compel a corresponding decrease in the breadth of it. Or we think of the way in which any variation in the slope of a ladder will cause the height that it reaches, to vary. Suppose we have got two such variables that depend one on the other. An alteration in one will bring about an alteration in the other, because of this dependence. Let us call one of the variables x, and the
