CALCULUS MADE EASY - Gutenberg

1 The Project Gutenberg EBook of CALCULUS Made Easy, by Silvanus ThompsonThis eBook is for the use of anyone anywhere in the United States andmost other parts of the world at no cost and with almost no restrictionswhatsoever. You may copy it, give it away or re-use it under the termsof the Project Gutenberg License included with this eBook or online If you are not located in the United States, youwill have to check the laws of the country where you are located beforeusing this : CALCULUS Made EasyBeing a very-simplest introduction to those beautifulmethods which are generally called by the terrifying namesof the DifferentiaAuthor: Silvanus ThompsonRelease Date: October 9, 2012 [eBook #33283]Most recently updated: November 18, 2021 Language: EnglishCharacter set encoding.

2 UTF-8** START OF THIS PROJECT Gutenberg EBOOK CALCULUS MADE EASY **transcriber s noteMinor presentational changes, and minor typographical andnumerical corrections, have been made without comment. Alltextual changes are detailed in the LATEX source PDF file is optimized for screen viewing, but may easily berecompiled for printing. Please see the preamble of the LATEX source file for MADE EASYMACMILLAN AND CO.,LimitedLONDON : BOMBAY : CALCUTTAMELBOURNETHE MACMILLAN COMPANYNEW YORK : BOSTON : CHICAGODALLAS : SAN FRANCISCOTHE MACMILLAN CO. OF CANADA, MADE EASY:BEING A VERY-SIMPLEST INTRODUCTION TOTHOSE BEAUTIFUL METHODS OF RECKONINGWHICH ARE GENERALLY CALLED BY THETERRIFYING NAMES OF THEDIFFERENTIAL CALCULUSAND THEINTEGRAL R.

3 EDITION, ENLARGEDMACMILLAN AND CO., LIMITEDST. MARTIN S STREET, Edition 1911 (twice), 1912, Edition one fool can do, another can.(Ancient Simian Proverb.)PREFACE TO THE SECOND success of this work has led the author to add a con-siderable number of worked examples and exercises. Advantage hasalso been taken to enlarge certain parts where experience showed thatfurther explanations would be author acknowledges with gratitude many valuable suggestionsand letters received from teachers, students, and , .. deliver you from the Preliminary Different Degrees of Smallness .. Relative Growings.. Cases .. Stage. What to do with Constants .. , Differences, Products and Quotients.

4 Differentiation .. Time Varies .. a Useful Dodge .. Meaning of Differentiation.. and Minima.. of Curves .. Useful Dodges .. true Compound Interest and the Law of Or-ganic Growth ..131viiCALCULUS MADE to deal with Sines and Cosines .. Differentiation .. as the Reverse of Finding Areas by Integrating .. , Pitfalls, and Triumphs .. some Solutions ..232 Table of Standard Forms ..249 Answers to Exercises.. many fools can calculate, it is surprising that itshould be thought either a difficult or a tedious task for any other foolto learn how to master the same CALCULUS -tricks are quite easy. Some are enormously fools who write the textbooks of advanced mathematics and theyare mostly clever fools seldom take the trouble to show you how easythe easy calculations are.

5 On the contrary, they seem to desire toimpress you with their tremendous cleverness by going about it in themost difficult myself a remarkably stupid fellow, I have had to unteachmyself the difficulties, and now beg to present to my fellow fools theparts that are not hard. Master these thoroughly, and the rest willfollow. What one fool can do, another DELIVER YOU FROM THE terror, which chokes off most fifth-form boys fromeven attempting to learn how to calculate, can be abolished once forall by simply stating what is the meaning in common-sense terms ofthe two principal symbols that are used in dreadful symbols are:(1)dwhich merely means a little bit of.

6 Thusdxmeans a little bit ofx; ordumeans a little bit ofu. Or-dinary mathematicians think it more polite to say an element of, instead of a little bit of. Just as you please. But you will find thatthese little bits (or elements) may be considered to be indefinitely small.(2)Zwhich is merely a longS, and may be called (if you like) thesum of. ThusZdxmeans the sum of all the little bits ofx; orZdtmeansthe sum of all the little bits oft. Ordinary mathematicians call thissymbol the integral of. Now any fool can see that ifxis consideredas made up of a lot of little bits, each of which is calleddx, if youadd them all up together you get the sum of all thedx s, (which is theCALCULUS MADE EASY2same thing as the whole ofx).

7 The word integral simply means thewhole. If you think of the duration of time for one hour, you may (ifyou like) think of it as cut up into 3600 little bits called seconds. Thewhole of the 3600 little bits added up together make one you see an expression that begins with this terrifying sym-bol, you will henceforth know that it is put there merely to give youinstructions that you are now to perform the operation (if you can) oftotalling up all the little bits that are indicated by the symbols s DIFFERENT DEGREES OF find that in our processes of calculation we have to deal withsmall quantities of various degrees of shall have also to learn under what circumstances we may con-sider small quantities to be so minute that we may omit them fromconsideration.

8 Everything depends upon relative we fix any rules let us think of some familiar cases. Thereare 60 minutes in the hour, 24 hours in the day, 7 days in the are therefore 1440 minutes in the day and 10080 minutes in 1 minute is a very small quantity of time compared witha whole week. Indeed, our forefathers considered it small as com-pared with an hour, and called it one min`ute, meaning a minutefraction namely one sixtieth of an hour. When they came to re-quire still smaller subdivisions of time, they divided each minute into60 still smaller parts, which, in Queen Elizabeth s days, they called second min`utes ( quantities of the second order of minute-ness).

9 Nowadays we call these small quantities of the second order ofsmallness seconds. But few people knowwhythey are so if one minute is so small as compared with a whole day, howCALCULUS MADE EASY4much smaller by comparison is one second!Again, think of a farthing as compared with a sovereign: it is barelyworth more than11000part. A farthing more or less is of precious littleimportance compared with a sovereign: it may certainly be regardedas asmallquantity. But compare a farthing with 1000: relatively tothis greater sum, the farthing is of no more importance than11000of afarthing would be to a sovereign. Even a golden sovereign is relativelya negligible quantity in the wealth of a if we fix upon any numerical fraction as constituting the pro-portion which for any purpose we call relatively small, we can easilystate other fractions of a higher degree of smallness.

10 Thus if, for thepurpose of time,160be called asmallfraction, then160of160(being asmallfraction of asmallfraction) may be regarded as asmall quantityof the second orderof smallness.*Or, if for any purpose we were to take 1 per cent. ( ) as asmallfraction, then 1 per cent. of 1 per cent. ( ,000) would be asmall fraction of the second order of smallness; and11,000,000would bea small fraction of the third order of smallness, being 1 per cent. of1 per cent. of 1 per , suppose that for some very precise purpose we should regard11,000,000as small. Thus, if a first-rate chronometer is not to loseor gain more than half a minute in a year, it must keep time withan accuracy of 1 part in 1,051,200.