Page:The New International Encyclopædia 1st ed. v. 04.djvu/31

CALCTTLirS. 17 great or small dt U supposed to be. The ratio jj- is called the differential coefficient of / with respect to t — the distance I being of course 'a function' of the time f. This ratio represents a limit. For. considering again the ball thrown up in the air, the error introduced by choosing a finite instead of an infinitesimal interval of time Is the less the smaller an interval is chosen, and finally the true velocity — is approached as a limit, when the interval of time becomes infinite- ly small. All this is concisely represented by a few symbols, as follows: limit /A/ _ dl_ &.t=0 M ) ~ dt In this expression A/ stands for some finite interval ('increment') of time, and AZ for the distance actually traversed during that intenal. And the expression tells that when Af ap- proaches zero (Af = 0) . ie. when it becomes in- A/ finitelv small, the ratio x; approaches as a limit CALCULUS. It need hardiv be remarked that the value -7-. dt while dl and dt are themselves infinitesimal quantities, their ratio may have any finite value, large or small. dl Maxima and Minima. — Since -p represents the dt velocity of the ball at any moment of the flight, it is evidently itself a variable quantity. For when, say, a rubber ball is thrown up in the air, the velocity of its motion becomes smaller and smaller imtil tlio highest point in its flight is reached; at that pjint the ball pauses for an instant and then begins to descend with increas- ing speed until it reaches the ground. Here it pauses again for an instant and then again goes up in the air. At the instant the ball is at the highest point, as well as at the instant it touches tne gi"Ound, the velocity is therefore zero : i.e. — = 0. But as the two points reached by the ball are respectively the highest and the lowest, it may be said that when the function I has its maximum or minimum value, its differential coefficient with respect to its variable (i.e, -p) is zero. This must be carefully remembered. Bearing in mind the ideas explained in the preceding paragraphs, the problems cited at the beginning of the article may now >e analyzed without any difficulty. I. Hohition of the First Problem. — In the prob- lem of the maximum rectangle, let a be the known sum of two adjacent sides, let x be one of the sides, and let y be the area of the rectangle. Then y ^ X {a — x) . or y ^ ax — a-". Seizing the rectangle at some point in its variation, let us lengthen the side x by some finite amount, Ar, and suppose that this causes the area to increase by a finite amount. Ay. Our equation then becomes 1/+ A ii=:a ( T-'- Ax) — (.r-^ Ax) "= ax+ax — T'—2rAx — ( A Jl*. Subtracting the original eqviation, y = ax — x', we get S,u=^a At — 2jrAr — ( Ax)' , and, dividing throughout bv Ax, A" - o ' , -7— =0— 2x— Ax. JIaking Ax smaller and smaller without limit, it will ultimately approach zero. Then a — 2^ — Ax will become simply a — ix, while the ratio -~- will approach its limit -^, and hence we will ax da- have dy Tx =<'-2-- Xow, it was shown above that, at the instant a function passes through its maximum value, its differential coefficient is zero. Hence, when the area of our rectangle is the greatest possible, dy ^— = 0, and therefore, a — 2a; = 0, or ax X = ^20. But this tells us that each side must be one- half of the known sum, i.e. that the two adjoin- ing sides must be made equal in order that the rectangle may have its maximum area. The process just employed in solving the prob- lem may be described as "differentiating with the aid of the theory of limits.' Indeed, we started with the law that the area of a rectangle equals the product of two adjoining sides, a law ex- liressed in our case by the equation y = xia — x) . We then ascertained the ratio of the finite incre- ment of area to an actual finite increment of the variable side x. Next we ascertained the limiting value of that ratio corresponding to an infinitely small increase of the side. This gave the value of the differential coefficient -^ of our dx function as a — 2cr. And as it had been shown before that the differential coefficient is zero at the point where a function has its maximum value, we wrote a — 2a; = 0, which gave the value of the side or for that point. By analogous processes of reasoning we may 'differentiate' any function whatever, and thus determine the form of its differential coefficient. In practical work, however, it is not necessary to go through the whole process every time a func- tion is differentiated, and the differential coeffi- cient of a fimction is usually obtained directly by the use of a few general formulas, the demon- strations of which are given in all text-books of calculus. In solving our other problems we will make direct use of two such formulas. II. Solution of the Second Pro1)lem. — In the problem of the person in a boat, call A the point --— ' 'J ^ -- " 3 "i^^-^^"^ 5-x where he must land in order to reach his point in the least time. An inspection of the accompanying figure shows that the distance of the boat from the point A is V 3- + x^ (hypothenuse of the right-angled triangle). To row this distance at the rate of 4 miles an hour requires |/ 3" -|- jr 4 (9 4- -rt i hours, or, as it may be written, -^ — — — — 4 hours. The distance of the landing point from the point of destination is denoted by 5 — x. To walk this distance at the rate of 5 miles an hour requires a- hours. The total time (call it