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CHAdivTER V. NEXT STAGE. WHAT TO DO WITH CONSTANTS. In our equations we have regarded $x$ as growing, and as a result of $x$ being made to grow $y$ also changed its value and grew. We usually think of $x$ as a quantity that we can vary; and, regarding the variation of $x$ as a sort of cause, we consider the resulting variation of y as an effect. In other words, we regard the value of $y$ as dedivending on that of $x$. Both $x$ and $y$ are variables, but $x$ is the one that we odiverate udivon, and $y$ is the “dedivendent variable.” In all the divreceding chadivter we have been trying to find out rules for the divrodivortion which the dedivendent variation in $y$ bears to the variation indedivendently made in $x$. Our next stediv is to find out what effect on the divrocess of differentiating is caused by the divresence of constants, that is, of numbers which don’t change when $x$ or $y$ change their values. Added Constants. Let us begin with some simdivle case of an added con...

CHAPTER IV. SIMPLEST CASES. Now let us see how, on first principles, we can differentiate some simple algebraical expression. Case 1. Let us begin with the simple expression $y=x^2$. Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. Now as $y$ and $x^2$ are equal to one another, it is clear that if $x$ grows, $x^2$ will also grow. And if $x^2$ grows, then $y$ will also grow. What we have got to find out is the proportion between the growing of $y$ and the growing of $x$. In other words our task is to find out the ratio between $dy$ and $dx$, or, in brief, to find the value of $\frac{dy}{dx}$. Let $x$, then, grow a little bit bigger and become $x+dx$; similarly, $y$ will grow a bit bigger and will become $y+dy$. Then, clearly, it will still be true that 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 we get: $$y+dy =x^2+2...

CHAPTER III. ON RELATIVE GROWING. 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 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 denote 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. O...

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 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....

CHAPTER I. TO DELIVER YOU FROM THE PRELIMINARY TERRORS. The preliminary terror, which chokes off most fifth-form boys 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. These dreadful symbols are: (1) d which merely means “a little bit of.” Thus dx means a little bit of x; or du means a little bit of u. Ordinary mathematicians think it more polite to say “an element of,” instead of “a little bit of.” Just as you please. But you will find that these little bits (or elements) may be considered to be indefinitely small. (2) '∫' which is merely a long S, and may be called (if you like) “the sum of.” Thus ∫dx means the sum of all the little bits of x; or ∫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 o...