Calculus Made Easy

"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."

"The preliminary terror, which chokes off most [students]... from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common sense terms—of the two principal symbols that are used in calculating."

"(1) d... merely means "a little bit of." Thus dx means a little bit of x or du means a little bit of u. ...you will find that these little bits or elements may be considered to be indefinitely small."

"(2) \int which is merely a long S... may be called, if you like, "the sum of." Thus \int dx means the sum of all the little bits of x or \int dt means the sum of all the little bits of t. Ordinary mathematicians call this symbol "the integral of.""

"Now any fool can see that if x is considered as made up of a lot of little bits, each of which is called dx, if you add them all up together you get the sum of all the dxs (which is the same thing as the whole of x). The word "integral" simply means "the whole.""

"When you see an expression that begins with this terrifying symbol, 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."

"We shall have... to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness."

"1 minute is a very small quantity of time compared with a whole week, Indeed, our forefathers considered it small as compared with an hour, and called it "one minùte," meaning a minute fraction—namely one sixtieth—of an hour. When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth's days, they called "second minùtes" (i.e. small quantities of the second order of minuteness). Nowadays we call these... "seconds." But few people know why they are so called."

"If, for the purpose of time, 1/60 be called a small fraction, then 1/60 of 1/60 (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness."

"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."

"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\cdot dx or x^2 dx or a^x dx are negligible. But dx\times dx would be negligible, being a small quantity of the second order."

"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\cdot 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 x2."

"The witty Dean Swift once wrote: "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 infimitum. 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."

"Let x and y be respectively the base and the height of a right-angled triangle (Fig. 4), of which the slope... is fixed at 30º. If we suppose this triangle to expand and yet keep its angles the same as at first, then, when the base grows so as to become x + dx the height becomes y + dy. Here, increasing x results in an increase of y. The little triangle, the height of which is dy and the base of which is dx is similar to the original triangle; and it is obvious that the value of the ratio dy/dx is the same as that of the ratio y/x."

"We call the ratio dy/dx "the differential coefficient of y with respect to x." It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ratio dy/dx."

"The process of finding the value of dy/dx is called "differentiating." But, remember, what is wanted is the value of this ratio when both dy and dx are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute."

"Let us begin with the simple expression y = x2 . ..Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. ...the enlarged y will be equal to the square of the enlarged x. Writing this down we have y + dy = (x + dx)2. Doing the squaring [see Ch.2, Fig.2 above] we get y + dy = x^2 + 2x\cdot dx + (dx)^2. ...dx2 will mean a little bit of a little bit of x; that is... a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have: y + dy = x^2 + 2x\cdot dx. Now y = x2; so let us subtract this from the equation and we have left dy = 2x\cdot dx. Dividing across by dx, we find dy/dx = 2x."

"Try differentiating y = x3. We let y grow to y + dy while x grows to x + dx. Then we have y + dy = (x + dx)3. ...[By a similar argument as above] dy/dx = 3x^2. ...Try differentiating y = x4. Starting as before by letting both y and x grow a bit, we have: y + dy = (x + dx)4. ...[By a similar argument as above] dy/dx = 4x^3."

"Let us collect the results to see if we can infer any general rule..."

"Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of x by 1 (for example in the last case reducing x4 to x3), and at the same time multiplying by a number (the same number in fact which originally appeared as the power). Now, when you have once seen this, you might easily conjecture how the others will run. You would expect that differentiating x5 would give 5x4 or differentiating x6 would give 6x5."

"Following out logically our observation, we should conclude that if we want to deal with any higher power,—call it n—we could tackle it in the same way. Let y = x'n, then we should expect to find that dy/dx = nx(n-1). For example let n = 8 then y = x8; and differentiating it would give dy/dx = 8x7."

"Thirdly, among the dreadful things they will say about "So Easy" is this: that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not? You don't forbid the use of a watch to every person who does not know how to make one? You don't object to the musician playing on a violin that he has not himself constructed. You don't teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus."