UC-NRLF $B SEfi 7Tb i«0?^ ^INFINITESIMAL ^LCULUS— FISHEF 1 UN BERKELIY BRARY UNIVERSITY OF V CALIFORNIA I Digitized by the Internet Arciiive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/briefintroductioOOfishrich A BRIEF INTRODUCTION TO THE INFINITESIMAL CALCULUS j2^^ A BRIEF INTRODUCTION TO THE Infinitesimal Calculus DESIGNED ESPECIALLY TO AID IN READING MATHEMATICAL ECONOMICS AND STATISTICS BY IRVING FISHER, Ph.D. Professor of Political Economy in Yale University Co-author op- Phillips's and Fisher's " Elements of Geometry' THIRD EDITION THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., Lxa I92I All rights reserved Copyright, 1897, By the MACMILLAN COMPANY. Set up and electrotyped 1897. Reprinted April, igooi July, 1901; February, 1904; March, 1906; March, 1909; September, 1916. Notisooti l^rees 1. 8. Gushing & Co. — Berwick & Smith Norwood Mass. U.S.A. Add'l 0^505 PREFACE This little volume contains the substance of lectures by which I have been accustomed to introduce the more advanced of my students to a course in modern economic theory. I could find no text-book sufficiently brief for my purpose, nor one which distributed the emphasis in the desired manner. My object, however, in preparing my notes for publication has not been principally to provide a book for classroom use. It must be admitted that very few teachers of Economics as yet desire to address their stu- dents in the mathematical tongue. I have had in mind not so much the classroom as the study. Teachers and students alike, however little they care about the mathematical medium for their own ideas, are growing to feel the need of it in order to understand the ideas of others. I have fre- quently received inquiries, as doubtless have other teachers, for some book which would enable a person without special mathematical training or aptitude to understand the works of Jevons, Walras, Marshall, or Pareto, or the mathematical articles constantly appearing in the Economic Journal^ the Journal of the Royal Statistical Society, the Giornale degli Economisti, and elsewhere. It is such a book that I have tried to write. S81 vi PREFACE The immediate occasion for its publication is the appear- ance in English of Cournot's Principes mathematiques de la theorie des richesses^ in Professor Ashley's series of " Eco- nomic Classics." The " non- mathematical " reader can only expect to understand the general trend of reasoning in this masterly Httle memoir. If he finds it as stimulating as most readers have, he will want to comprehend its notation and processes in detail. I have tried in some measure to meet the varying needs of different readers by using two sorts of type. If desired, most of the fine print may be omitted on first reading, and all on second. The reader is, however, advised not to pass over all of the examples. Although intended primarily for economic students, the book is equally adapted to the use of those who wish a short course in " The Calculus " as a matter of general education. I therefore venture the hope that teachers of mathematics may find it useful as a text-book in courses planned espe- cially for the " general student." I have long been of the opinion that the fundamental conceptions and processes of the Infinitesimal Calculus are of greater educational value than those of Analytical Geometry or Trigonometry, which at present find a conspicuous place in our school and college curricula. Moreover, they are almost as easily learned, and far less easily forgotten. IRVING. FISHER. New Haven, September, 1897. PREFACE TO THE THIRD EDITION In the present edition have been incorporated several changes and additions originally prepared for the German translation of 1904 and for a Japanese translation in prep- aration. A preliminary statement of the concepts of limits and several new examples have also been inserted. IRVING FISHER. November, 1905. CONTENTS PAGB Introduction xi CHAPTER I The General Method of Differentiation . . , , i CHAPTER II General Theorems of Differentiation . . . . i6 CHAPTER III Differentiation of the Elementary Functions ... 30 CHAPTER IV Successive Differentiation — Maxima and Minima . . 37 CHAPTER V Taylor's Theorem 49 CHAPTER VI Integral Calculus 57 APPENDIX Functions of More than One Variable . . . .73 INTRODUCTION The reader of the following book should be familiar with ordinary algebraic operations and with the concepts of vari- ation and limits, a brief statement of which is here appended. Continuous Variation. — Suppose the line ab to represent all possible magnitudes between — a and -f b ; suppose om to represent one magnitude between — a and + b ; this magnitude is said to vary continuously when it increases or —-a o m m\ nity mo + d Z i L, I I 1 Fig. I. decreases in such a manner that m may occupy any position whatever between — a and -f b. Limits. — If we conceive om to have an infinite succes^on of magnitudes such that /// may occupy the positions nii, Wj, Wg, etc., making the ultimate difference between ob and om less than any assignable positive quantity, then om is a vari- able and ob is its limit It is clear, then, that the difference between the limit ob and the variable om is another variable magnitude whose limit is zero. A variable, with a limit zero, is called an infinitesimal. xii INTRODUCTION Application to Infinite Series. — In a converging infinite series, the sum of each successive term and those preceding approaches a magnitude understood to be designated by the series. This magnitude is called the * sum ' of the series. Thus, the repeating decimal .666 •••, ID lO^ lO^ ID* means a series of successive magnitudes, viz.: {a) — , which is less than |. {b) 1 2> which is less than |, but more nearly approxi- mates I than {a), (c) 1 sH 5, which is less than |, but more nearly approximates | than (^). (d) 1 r. H 7, H 7, which is less than |, but more ^ ' lO lo-^ lO^ lO^ ^' nearly approximates | than {c). Thus, as the number of terms of the series is increased, the sum of the terms remains always less than |, but approx- imates ultimately as nearly | as may be desired, i.e. converges towards |. We therefore, by convention, speak of J as the ' sum,' or limit, of this infinite series. Theorems. I. The limit of the sum of two different variables {which approach limits) is the sum of the limits of those variables. INTRODUCTION xiX 2. The limit of the difference of two different variables {which approach limits) is the difference of the limits of those variables. 3. The limit of the product of two different variables {which approach limits) is the product of the limits of those variables. 4. The limit of the quotient of two different variables {which approach limits) is the quotient of the limits oj those variables. INFINITESIMAL CALCULUS CHAPTER I THE GENERAL METHOD OF DIFFERENTIATION 1. The Infinitesimal Calculus treats of the ultimate ratios of vanishing quantities. This definition, however, can only become intelligible after some actual acquaintance with "ultimate ratios." 2. The conception of a limiting or ultimate ratio is funda- mental in many familiar relations. It is impossible, without it, to obtain a clear notion of what is the velocity of a body at an instant. The average velocity of the body during a period oi time may readily be defined as the quotient of the space traversed during that period divided by the time of traversing it. If a steamer crosses the Atlantic (3000 miles) in 6 days, we may say that the average speed is 3000 -i- 6, or 500, miles per day. But this does not tell us the speed at various points in the voyage, under head winds, storms, or other conditions, favorable or unfavorable. What, for instance, was the speed at noon of the third day out? We may obtain a first approximation to the desired result by taking the average speed for a short time after the given instant ; that is, taking the ratio of the distance traversed 2 INFINITESIMAL CALCULUS during (say) the following hour to the time of traversing it, which is -^-^ of a day. If this distance be 20 miles, we obtain 20 -7- -Jj, or 480 miles per day, as the average speed during that hour. For a second approximation we take a minute instead of an hour ; for a third, a second instead of a minute, and so on. The ratio of the space traversed to the time of traversing it becomes closer and closer to the true speed. Though both the time and space approach zero as limit, their ratio does not. The limit which this ratio approaches, or the ultimate ratio of the distance traversed to the time of traversing it when both distance and time vanish, is the pre- cise speed at the instant. 3. Let us apply this method of obtaining velocity to bodies falling in a vacuum. We know from experience that the distance fallen equals sixteen times the square of the time of falling, i.e. s= 16/^, where s is the distance fallen from rest (measured in feet), and / is the time of faUing (in seconds). Consider the body at some particular instant, / being the time to this particular point and s the distance. Suppose we wait until the time has increased by a small increment A/, during which the body increases its distance from the starting-point, s^ by the small increment Aj". Since the above formula holds true of all points, it holds true now, when the time is t-\-l^t, and the distance is j-f-Aj. That is, s + As= 16 (t + Aty. This gives j + Ai-= 16/2+32/'. AZ-h i6(My. But s =i6t\ Subtracting, we have A.f=32/. A/+ i6(A/)2, li GENERAL METHOD OF DIFFERENTIATION 3 whence — *^ = 3 2 ^ + 1 6 A/. (i) This is the average velocity during the small interval A/. Thus, if A/ = I second and / be 5 seconds, the average speed of the body during that half second (viz., the one beginning 5 seconds from rest) is 32 X 5 + 1 6 X ^, or 1 68 feet per second. If we take ^\^ of a sec- ond instead of \, we have 32 x 5 + 16 X y^^, or 1 60.1 feet per second. Thus, by taking A/ smaller and smaller, we obtain the Ax average velocity — for a smaller and smaller interval of time A/ immediately after the completion of the fifth second. The Aj" limit which — approaches, as A/ approaches zero as its A/ limit, is called the velocity at the very instant of completing the fifth second. Its value is exactly 160, as is evident from the right-hand member of equation (i), which approaches as its limit (as / is 5 and A/ approaches zero), 32 X 5 + 16 X o, or 160. In general, to express the limit of both sides of equation (i) when A/ approaches zero, we write lim — = '?2 /. A/ ^ 4. The student will observe that, as A/ approaches zero, Aj also approaches zero, since a body cannot pass over any distance in no time. He must be warned:> however, against expressing the limit of — by -, which, of course,- is quite indeterminate. But in spite of the fact that the ratio of these limits of Ai and A/ is indeterminate, the limit of the ratio of Ax and A/ 4 INFINITESIMAL CALCULUS may be entirely determinate. It is only with this latter con- ception, viz. the limit of — , or lim — , that the student has to deal. ^' ^' The Hmit of the ratio of the vanishing quantities Aj- and Ai" A/, or lim — , is called the " derivative " of j with respect to ^* As t ; because, from s= 16 /^ we denve hm — = -12 f. A/ ^ In fact, we may speak of either member of the latter of these two equations as the derivative of either member of the former equation. For instance, 32 / is the derivative of 16 /^. 5. Other names and notations are also used. Thus in- stead of lim — it is usual to employ the shorter symbol — . A/ ^ ^ ^ dt In this expression ds and dt are called differentials of s and /, just as Ai" and A/ are called increments of s and /. But they are not zeros. They have no definite value individually. We may select any value we please for one of them. But when this one is fixed, the other is also, since the two must be kept in a ratio equal to lim — . We say therefore that the differ- entials ds and dt are any two quantities which bear to each other the ratio which is the limit of the ratio between Aj and A/. Other names for lim — or —.besides "derivative," are A/ dt' " differential quotient " and " differential coefficient." 6. In the particular case considered above, the differ- ential quotient is a velocity and may be denoted by v. Equation (2) thus becomes* v=T,2t. * If distance be measured in centimetres instead of in feet, we should have V = 980 /, and in general v =gty where ^ is a constant depending for its numerical value on the units chosen for measuring space and time. GENERAL METHOD OF DIFFERENTIATION 5 Velocity at a point may now be defined as the ultimate ratio of the space traversed Just after passing the point to the time of traversing it when the space and time approach zero as limit. 7. Examples. 1. What is the velocity of a body which has fallen 10 seconds ? 100 seconds ? i^ seconds ? 2. What is the velocity of a body which has fallen 16 feet ? Hint. — First find how many seconds it has fallen by using j=i6/-. 3. What is the velocity of a body which has fallen 64 feet ? 4 feet ? I foot ? 2 feet ? 4. It being known that a body, falling not from rest, but with an initial velocity of 5 feet per second, obeys the law s= 16/2+5/, (l) what will be its velocity at the end of any time t ? Hint. — Let / receive an increment A/, causing s to increase by ^s^ so that s-\-£^s= l6(/ + A/)2 + 5(/ + A/). (2) Subtract (i) from (2), divide by A/ and then reduce A/ and \s to zero. Ac Ans. lim - = ^2/ 4- C. A/ "^ ^ 5. What will be the velocity at the end of 10 seconds? At the end of 69 feet ? 6. It being known that a body falling with an initial velocitv of u obeys the law s = \gt'^ + m^i what will be its velocity at the end of time/? When/=3? . 8. When one quantity depends upon another, the first is said to be a function of the second. A change in the second is in general accompanied by a change in the first. In each case the limits, within which the function relation exists should be specified. 6 INFINITESIMAL CALCULUS Thus the distance a body falls from rest is a function of the time of falling, for how far the body falls depends on how long it has fallen; - the demand for an article is a function of its price, for if the price changes the demand changes; \i y — x^, then jj/ is a function of jr, for a variation in the magnitude of x necessitates also a variation in the magnitude of y. 9. When one quantity is a function of another, the latter is called the independent variable^ and the former the de- pendent variable. The distinction between the independent and the depend- ent variable is only for convenience of expression. The two may be interchanged. Thus, as the distance of a falling body from the starting-point changes, there is also a change in the time it has taken. Hence we may say that " time of falling " is a function of " distance fallen." Simi- larly price may be regarded as a function of demand. Again, y = x'^ may be written x = Vy, thus making x a function of y. The idea of functional dependence is therefore quite different from that of causal dependence. Functional dependence is a wm/m^/ relation. In the example of falling bodies s was a function of /, and what we accomplished was to find the differential quotient or derivative of that function. The derivative in this case was a velocity. In general the process of finding the differ- ential quotient of any given function is called differentiation^ and is the subject matter of the Differential Calculus, one of the two branches into which the Infinitesimal Calculus is divided. The Differential Calculus will occupy us in the first five chapters of this book. 10. A second important application of the idea of a differ- ential quotient of a function is to the tangential direction of a curve at any point on it. The Calculus enables us to conceive in the most general manner of a tangent to a curve. The GENERAL METHOD OF DIFFERENTIATION 7 Student should observe that the usual definition of a tangent to a circle will not apply to any and all curves. A straight line may have only one point in common with a curve and yet cut it and not be tangent. 1 1 . Let RS be a curve whose equation is y=^\^-^x-x^, (i) That is, for atiy point P upon it, the " ordinate," y (or dis- tnnrp^ P4 from that point to the horizontal axis), is related Fig. I. to the ** ahsrj <;sa^''WW ^l^gfonr^^^ Q^ from the vertical axis), in the manner expressed by (i). PA is a function of OA ; i.e. the height, PA, of any point P on the curve depends upon its distance, OA, from the vertical axis. What is the direction of the curve at the point P ? The direction from the point P to another point P is the direc- tion of the secant line QPP. The point P has for abscissa, 8 INFINITESIMAL CALCULUS X 4- A^, and for ordinate, y + A>'. Since the relation (i) holds true of all points on the curve, it holds true of P . Hence j + Ay = i + 5 (jv + A^) — {x -{- Ajc)^, or ^H- Aj' =1 + 5^ + 5 A^ — ^— 2:vAj£:— {J^ocf. Subtracting y=. i -\- ^x — 0^, we have Aj = 5 ^x — 2 x ^x — (Ajv)^, whence — = k — 2 x — Ajc. Ax ^ We may pause here a moment to see what this result means. ^^ or — is the " slope " of the line Q'FP'. That Ax PC ^ ^ is, it is the rate at which a point moving from Q' toward P rises in proportion to its horizontal progress. It is the same sort of magnitude as that referred to as the " grade " of an uphill road which rises " so many feet to the mile (hori- zontally)." If -^ = — , QPP rises one foot in every ten horizontally. The " slope " of a line shows its direction. At' The equation -^ = ^ — 2x — ^x shows that the "slope" of the secant line Q'PP' is to be found by taking 5 and subtracting, first, two times the number of units in OA and then the number of units in AB. For instance, if OA = 2 and AB = ^, then -^= 5 — 2x2 — A = *; t.g. the secant slopes i foot up for every 2 feet sidewise. 12. But we have not yet reached the tangent at P. Let the point P' be gradually shifted along the curve toward P until it ultimately coincides. The secant QP' will gradually GENERAL METHOD OF DIFFERENTIATION 9 change its direction and approach a limiting position QP, This limiting position we call the tangent. Its slope is dy Thus, if x(i.e. OA) is 2, ^= i. That is, QP is inclined at 45^ dv If 4; is 4, -^ = — 3 ; i.e. the curve slopes down^ not up. Fig. 2. —A, positive slope; B, zero slope; C, negative slope. Examples. 1. What is the slope of the tangent to the above curve at the point whose abscissa is i ? o ? 2J ? What does the answer to the last mean ? 3 ? What does this mean ? 6 ? — i ? 2. Derive the formula for the slope of the tangent to the curve y — \ -\- X -\- x^. 13. To construct a tangent at P, all we need to do is to draw a Une through /'with the required slope. Thus, if we wish the tangent to the point whose abscissa is i, we find from the above formula that its slope is 3. We therefore lay off a horizontal line LM (Fig. i) equal to any length ^jc, and at its extremity erect a vertical, MN, equal to three times as much, or dy. Draw LN \ this has the required direction. Then through P draw a line parallel to LN. This will be the tangent. We may also call PC, dx and /"'C, dy, for, by Sec. 5, dx and dy are simply any two magnitudes having a ratio equal to the limit of -^ when A^ approaches zero as its limit. t^x The prublem of drawing a tangent and calculating its slope was one of the chief problems which gave rise to .the discovery of the Calculus. 10 INFINITESIMAL CALCULUS 14. It is evident that we could approach P from the left as well as from the right. We should, however, reach the same limiting posi- tion unless there should be an angle in the curve at the point /* as in Fig. 3. In this case, the progressive (/'A') and regressive {HP) tan- gents do not coincide. _^^_ 'K ?, Such peculiar points are not considered in this little treatise. All the functions are such that, for the values of the independent variable which are considered, the progressive and regressive derivatives are identical. The curves considered are all "smooth," that is, have.no angles or sudden changes in direction. In many applications of the Calculus, such as to statistical or economic diagrams, it is often con- venient first to smooth out the curves considered. When we want to see from a plot of the population what is the general rate of increase, we draw a tangent not to the plot of the actual figures, but to a smooth curve coinciding as nearly as possible with the plot. The student will be able to satisfy himself in every particular case to be considered that the progressive and regressive derivatives are identical. Thus, for j=i6/2 in section 3, let / receive a decrement A't, causing s to have a decrement a's. Then s - A's = i6(t - A'ty. Expanding, subtracting, and dividing as before, we obtain ^=32t-l6A't, A't which reduces at the limit to d's — = 32 /, as before. d't ^ Indeed, we assume in general, that it is physically impossible for a body to change its velocity />er saltum. Hence the definition of GENERAL METHOD OF DIFFERENTIATION 11 velocity given in section 6 is equivalent to the following alternative definition : the ultimate ratio of the space traversed just before reaching the point to the time of traversing it when the space and time ap- proach zero as limit. We shall, therefore, henceforth treat only of functions whose deriva- tives are continuous and which are themselves continuous, within the limits considered, that is, which in changing from one value to another, pass continuously through all intermediate values. 15, We have seen that the conception of an ultimate ratio clears up the notion of velocity in mechanics and tangential slope in geometry. It is also applicable to much else in both these sciences as well as in all mathematical sciences. Momentum, acceleration, force, horsepower, density, curva- ture, marginal utility, marginal cost, elasticity of demand, birth rate, " force of mortality," are all examples. The conception of an ultimate ratio or of the derivative of a function is not dependent, however, on any special appHca- tion. It is purely an abstract idea of number. 16. Thus let two variables x and y fulfil the equation ;; = x^, where « is a constant and a positive integer. We may obtain the differential quotient -^ for any particular value dx of X, as follows : Let X receive an increment Ajc producing an increment of y denoted by Ay. Then, by the binomial theorem, ^ + Aj = (^ -h Ajc)", 2 = Jt:** 4- WJC""^ A^ -I- Ajx:^ (•••). 12 INFINITESIMAL CALCULUS Subtracting y = x\ we have A); = «ji:"^AxH-(A^)2(...) Whence ^=«:c«-^ + A^ (...), ivhere the parenthesis is evidently a finite quantity and re- mains finite after Ax becomes zero. Hence, when Ajc: becomes zero, the term Ax(---) becomes zero, and the equation becomes, dx 17. This is the first and most important specific formula which we have reached for the derivative of a function. It states that, to obtain the derivative of .r", a power oi x, we need only reduce the exponent by unity and use the old exponent for coefficient. Thus the derivative of x^ is t^x^. When x passes through the value 2, 3^2 becomes 12; that is, j, or x^, is increasing I2 times as fast as x. -^ is the rate at which j increases compared with the rate we make x dx increase. If y denotes the distance of a moving body from the start- ing-point, and X denotes the time it has moved, y-, or 3 x'^, expresses its velocity. Again, if x and ^ are the "coordinates" {i.e. the "ab- scissa" and "ordinate") of a curve whose equation is ^ = ;f^, then 3 x'^ is its slope at the point whose abscissa is x. Although it is logically unnecessary, it is practically helpful to pict- ure the differential quotient as a possible velocity or a possible slope. Of the two independent discoverers of the Calculus, Newton seemed to have employed the former image, and Leibnitz the latter. New- ton's term for a differential quotient was " fluxion." Examples. — 1. Find the derivatives of ^^^^ ^^ ^2^ ^^ What is the meaning of the answer to the last ? 2. How many times as fast does y increase as x when y —x^ and A- is 2 ? 3. How fast does x^ increase compared with x when ^ is - i ? What dofs the negative answer mean ? GENERAL METHOD OF DIFFERENTIATION 13 1 8. The process employed in this chapter for obtaining the derivative of a function is called the *' general method of differentiation." It consists (i) in giving to the independent variable a small increment, thus causing another small incre- ment* in the dependent variable or function ; (2) in writing the relation between the two variables first without and then with these increments and subtracting the first from the second ; (3) in dividing through by the increment of the in- dependent variable ; (4) in passing over from -^ to ^. This process should be thoroughly mastered by the student, for it contains, in embryo, the whole of the Infini- tesimal Calculus. He will observe that the order of steps (3) and (4) cannot be inverted without producing the barren result = 0. 19. Nevertheless, we can anticipate the result of step (4) without changing from the form of (2). Thus, the equation yields at step (2) : A>'= 2 A^-i-6 ;c Ajt:-f-3 {J^xf-\- 15 x^ lx-\-\%x{p.xf-\-<^{p.xf ^(2 + 6^4-15 ^')^^' +(3 + 15 ^)(A^)' + 5 (M'- It can readily be foreseen that step (3) {i.e. dividing by ^x) will remove the first \x, and reduce the exponents of the powers of Ajc by one, and that therefore when step (4) is performed {i.e. reducing \x to zero), all terms beyond the first will disappear, leaving 2 + 6 jic + 15 -^ ^s the derivative. Now it is clear that this result could have been anticipated simply by neglecting the terms involving powers * Decrements may always be regarded as negative increments. 14 INFINITESIMAL CALCULUS of Ajc higher than the first, and taking the coefficient of the first power as the required derivative. Though this process of neglecting certain terms at step (2) is a mere anticipation of what must necessarily happen at step (4), it may be shown to be perfectly natural in situ. If A^ be less than one, (A^)- will be less than Ajc, and (A^)^ less than (Ajt:)^, etc. By making A^ smaller and smaller, the higher powers (Ajc)^, (A^)^, etc., can be made indefinitely small, not only absolutely, but in comparison with Ajc. The higher powers of AJt thus growing negligible relatively to Ajc, the terms in which those powers occur as factors must also grow negligible (provided, of course, the other factor composing each such term does not approach infinity as limit). Thus, if A^ is y^^, {Lxf is jir^, and (A;r)8 only xinnjW* Con- sequently in the equation Ay = (2 + 6 ^ + 15 0:2) A;^ -j- (3 + 15 ;r) (A^)2 + 5(A;r)3, we can, by reducing t^^ sufficiently, make the terms beyond the first as small as we please compared zuith the firsts no matter what be the value of x^ so long as it is finite, thus keeping the parentheses finite. For instance, if x be 2, we have A^ = 74 A;r + 33 (A^)'-^ + 5(Ajr)3, Then, if Lx be .01, this becomes Ar = .74 + .0033 + .000,005. If A^ = .001, it becomes t^y - .074 + .000,033 + .000,000,005. If A^ = .000,001, it becomes Aj = .000,074 + .000,000,000,033 + ,000,000,000,000,000,005, and the smaller we make A;r, the more negligible become the terms involving (A;r)2 and (Ajf)^, until at the limit they become, not simply negligible " for practical purposes," but absolutely negligible. GENERAL METHOD OF DIFFERENTIATION 15 The anticipatory neglect of terms involving powers of ^x higher than the first often saves a great deal of labor. Examples. 1. Find -^ when y = j^, dx 2. Find ^ when v = ;r7 + 8;t« + 4. dx 3. Find ^ when y — \ox^^, dx ^ 4. Find -^ when y = ax"^ + dx^, tn and n being constant and integral. ^ Ans. amx^-^ + dnx"-^. 5. If X, the side of a square, has an increment ?, what will be the increment of the area of the square ? 6. In the function y = ;^ x"^ + 2 x, find the value of x when y in- creases 20 times as fast as x. Ans. x = 3. Differentiate the following functions : 7. jj/ = 3 al^x^ + c. 8. y = 4x^ — yx^-\-2x^2a. Ans. 20 x^ — 21 x"^ -\- 2. 9. y = ^j(^-{a + b')x, 10. yz^^b-^-xY- bx\ Ans. z ^ -^ A bx ■\- z x^. 16 INI^INITESIMAL CALCULUS CHAPTER II GENERAL THEOREMS OF DIFFERENTUTION 20. If we differentiate y— 2X by the general method, we obtain Clearing this equation of fractions, we have dy=2dx. (2) This last equation is simply another form of the first, and more convenient for some purposes. Thus, dy — d xdx is a transformation of dx which in turn means lim — ^ = 6 ;r. ^x 6 ;f is a differential quotient and 6 xdx is a differential. These conceptions are strictly correlative. To obtain the differen- tial quotient from the differential, we simply divide by dx ; to obtain the reverse, we multiply by dx. GENERAL THEOREMS OF DIFFERENTIATION Vi Examples. 1. What is the differential of ;«^? 2. The differential quotients of ^r", x^"^, x*? 21. To express the mere fact that j is a function of x, without specifying exactly w/iaif function, it is customary to use the letters ^, /, <^, if/ (and rarely others) followed by x in a parenthesis. They may be regarded simply as abbrevia tions of the word " function." Thus j = Function of jc is abbreviated to y = J^(x) . It is to be observed that the letters F,/, 0, ^, etc., do not repre- sent quantities like x and ^, but, like A and £^, represent operations on quantities. 22. The general expression for a function, such as <^(^), is often used to express, within brief compass, any special function. Thus if we have the equation 1+^-6^ + — ^~ ^x 2 —x^ * we may shorten this to y={x) by denoting the clumsy right-hand member by . The differential oi F{x) is therefore expressed by F\x)dx. 24. Another meth' d of expressing the differential quo- tient of F(x) connects it with the general method of differentiation. Thus, if X receives an increment A^, Fix) will become F{x-\-^x), This differs from its original value F{pc) by F{x^t^x)-F{x)'. The ratio of this increment of the function to the incre- ment ^x^ of the independent variable x^ is F{x-\- ^x) — F(x) t^x - i. ., . r F(x-^b.x^- Fix) Its hmit, VIZ. hm — ^ — ^ ^-^, is the differential quotient of Fipc) ; i.e. is F\x). The above process is identical with the general method of differen- tiation, though we have expressed it without the use of y. We might have proceeded as follows : GENERAL THEOREMS OF DIFFERENTIATION 19 Put F^pc) equal to y so that . y = ^W. Subtract this from y -}- Ay = F(j: -{^•^Ax), ' ^ and divide by Ax^ giving \ Ay _ F{x + Lx^-r E{x) Ax~ Ax or, at the limit ± = lim ^(^ + ^^)-^(^) dx Ax 25. Yet one more notation should b' .miliarized. Rather it is a new application of ^n old one. Instead of writing -^, we may replace y ^ this expression by F(x)y so that it reads 'iJZMl. dx The student will do well now to release his mind from y as any necessary element in the analysis. It is to be regarded merely as a further abbreviation of F(x). F{x) rather than/ is to be thought of as primarily the function of (jf). Thus, in our introductory example, instead of denoting space by s and writing s = 16 i^, we need only say if / denotes time, the function of /, 16 /2^ will denote space. So also if X denotes the abscissa of a curve, F{x) instead of y de- notes its ordinate. Thus, ^^f!)is2A:, c/x or d(x^)=2xdx. Examples.- ^^ = ^ ^(^)=? We thus have five methods of denoting the differential quotient ofy, or its equal F{x) ; viz. : 20 INFINITESIMAL CALCULUS 26. If a function of x is the sum of several functions of X i.€, if then, since this equation holds true of all values of x, it holds true when x becomes x + Ajc, so that F{x + A^) =/i<^ + A^) 4-/2(^ + A^) + -. Subtracting the upper equation from the lower, and divid- ing by A^, we obtain Fix + Ay) - F{x) ^ /i (^ + A^) -/i {x) Aa: ^x ^ f,{x-^^x)-Mx) ^ Ax Now let Ax approach zero as its limit. Then for the limits of the terms in the above equation, we have : Ax Ax or . F'(x)=/,'(x)-\-/J(x)+,etc.' That is, fke differential quotient of the sum of several func- tions is the sum of the differential quotients of those functions. The same reasoning establishes the corresponding theorem for the difference of functions. Thus the differential quotient of ^r^ _}. ^ \^ ix-\- ^x^. Sometimes the theorem is used in the differential form F'(x)c/x=/i'(x)dx+/2'ixyx+-, or again F'^xyx = [/i'(^) 4-/2' W + .••] dx. Examples. — Find the differential quotient of: 1. x^ + x^-X^. 2. xT-x^ + x. 3. -x^-^x^^ GENERAL THEOREMS OF DIFFERENTIATION 21 27. If a function of x is the sum of another function of X and a constant quantity, i.e. if . F{x)=f{x)^K, (i) where ^ is a constant, then •F\x)=f{x\ (2) the same result as if K were not present in (i) at all. The proof of (2) is simple. When x becomes x^i^x, (i) be- comes F{x + A^) =/(^ + Ax) 4- K. (i)' When we subtract (i) from (i)', K disappears entirely, and we have, after dividing by A„t, F{x + Ajc) - F{x) _ fix + Ax) -f(x) Ax "" Ax ' which reduces at the limit to (2). The same result would be obtained if in (i) K were preceded by the minus instead of the plus sign. Hence, to obtain the derivative of the sum (or difference) of a series of terms, some of which are constants, we simply take the sum (or difference) of the derivatives of all the terms which are functions of x, ignoring those which are constant. Thus, if j^ = ^3 + 5^ ^ = 3;f2. Again, the derivative of x^-x* + x + a-d-S is s^-4^^+i' The foregoing result is sometimes expressed by regarding all the terms, even the constants, as functions of x, and saying that the derivative of a constant term is zero. 22 INFINITESIMAL CALCULUS Examples. — Find the differential quotient of: I. x^ + 2. 2. ^r'^ + 3 + x^. 3. xr^ ^jc'-V 19. 4. Prove last by general method of differentiation, 28. If a function of x is the product of a constant by another function of x^ i.e. if F{x) = K{x\ (i) then F\x) = K\x)', (2) that is, the derivative of the product of a constant by a func- tion is the product of the constant by the derivative of the function. Proof. — When x becomes x -\- Ax, (i) becomes F(x + Ax) = X(l>(x + Ax). (i)' Subtracting (i) from (i)' and dividing by Ax, we have F(x + Ax) - F{x) _ K4> (x 4- Ax) - X (x) Ax Ax j^ (x) ^ = A > Ax or at the limit, Fix) = K'(x). Corollary. — The derivative of mx"" is m times the de- rivative of ^% as given in § 16. Hence, it is mnx^~^. This result is so often used that it should be carefully memorized. When n\^\, the derivative is simply 771. (Show this directly, by § 18.) Examples. •iffere 5- :ntiate 4■^^^ 3^, \^. 3<^ 3' y/2x^ 5 x4i + - I V5 ' -V2, GENERAL THEOREMS OF DIFFERENTIATION 23 29. If a function of x is the product of two functions of X, i,e. if Fix) = (x){l/(x), then F(x + Ax) = (x-{-Ax)if/(x-{- Ax). Subtracting and dividing by Ax, we have F(x + Ax) — F{x) __ {x-\- Ax)if/{x + Ax) — (x) if/ (x) Ax Ax The right member may be changed in form without suf- fering any change in value by adding and subtracting <^ (x) i{/(x-\- Ax) in its numerator, giving 4>(x-\-Ax)^//(x-{-Ax)-<|>(x)^f^(x)-<|>(x)^|/(x-\-Ax)-]-(f>(x)^//(x+^x) Ax Grouping the terms according to common factors, we have [<^ (x + Ax)- (x)'] if/ (x -\-Ax) + (x)[i{/ (x + Ax) - if/jx)^ Ax ^ ' or Ax r\ " / ^^ Taking these terms in order, we see that the 1- v .(x-\-Ax) — (x) . ,. . hmit of -^ T^ — r^_z_ js f^'(x), Ax ^ limit of if/(x-\-Ax) isi{/(x), ,. . . il/ (x -{- Ax) — d/ (x) . ., , hmit of ^-^ — ■ — -^ ^-^^ is i^\x)f Ax limit of {x) is <^(^), which gives for the limit of the right member of the equation '(x)il,(x)-\-^{,\x)(x); 24 INFINITESIMAL CALCULUS while for the other (or left) member of the equation the ,. . F{x -\- t^x) — F{pc) . hmit of -^ -^ ^^ IS F'(x), Putting these limits equal, we have F\x) = Xx) if; (x) + ij;'(x) (x). In words, ^/le derivative of the product of two functions is the sum of the products obtained by multiplying the derivative of each function by the other function. dx dx dx = 2x(l-\-X^)+ 2X'X^ = 2jr(l + 2X^). Examples. — 1. Find the derivative of (i -\- x'^)(^i ~ x'^) first by § 29 and afterwards by multiplying out and then differentiating. 2. (2 + ^5 - ^)(5 + ^), 4(^2 + I.) (^3 _ 2), a{sx^ + 4)(Sx^+6x^ + 7x-\-S), {a + d) {kx^^^ hx^-\- p) {qx"^ + r). 3. Prove § 28 by using § 29, regarding i as a form of \^(;r)5 whose derivative is zero. (See § 27, end.) 4. Prove § 29, using a different notation. 30. Corollary. — If F{x^-=fx(x)f^{x)fz{x')i we may abbrevi- ^\.^ fiipc) fzix) to 0(jr), so that whence P{x) =fi'{x) 4>{x) + '(x)/i(x). Replacing (f>(x) by its value ^(;r)y^(.r) and (X)/,(X), we Kave F'ix) =/,'(x) [/2 W/3 W] + [/2'(X)MX) 4-/3' W/2(^)]/i(^) =/l'W/2W/3W +/2'W/3W/l(-^) +/3'W/lW/2(*). gENERAL THEOREMS OF DIFFERENTIATION 25 By successive applications of § 29 this theorem can be generalized to the product of any number of functions, and in words is as follows: The derivative of the product of any number of functions is the sum of. the products obtained by multiplying the derivative of each function by the product of all the other functions. Examples. — Find the derivatives of (.^2 +i){x+ i){x - I), .r3(^2 + 2x + 3)(2;«:4 - 7)(4 - ^). 31 • If ■^{^') — ^ . , and {x) is not zero, then Fix + Ajc) — F{pc) _ <^(x + Ajc) <^{x) ^x Ax _ (x) — {x + Ax) ~' Ax (fi{x)(j>{x-^Ax) _ ~^ (x) ~ {x)(x-\-Ax)' Ax or at the limit ^'W = -ft=-^2- <^'(^) l{x)j That is, the derivative of the reciprocal of a function is minus the derivative of the function divided by the square of the function. Thus the differential quotient of — - is 3^ -^(3^2) ^^ or -^, or - 2 (3;r2)2 9^* 3^ 32. Examples. 1. Find the derivative of ^' 1+^' l+x + X^ x^ (I+^)''^ (l+X + *= 26 INFINITESIMAL CALCULUS 2. Show by method of § 29, that if 'AW then J.. ^t!±j=Lf±^ where the (;r)'s are omitted for brevity. 3. Prove the same theorem by applying results of §§ 29, 31, after throwing — in the form — 33* We may interject here an application of the result of § 31 to generalizing the theorem of § 16. The differential quotient of x^ was there obtained only under the restriction that « be a positive integer. But if « be a negative integer, — m, then x"^ becomes — . This x^ fraction has meaning only provided the denominator is nut zero, i.e. x is not zero. The differential quotient becomes — mx^-^ X'^rr^ ' which reduces to — mx-^-^ or nx*^-^. That is, the restriction imposed in § 16 that n must be positive, may be removed. Examples. 1. Differentiate x'^. 2. Differentiate 3^"^ 3. Differentiate — . 4. Differentiate -^. x-i Sx^ 34. If we wish to differentiate the quotient of two func- tions as ^) ( , we can do this by combining the results of §§ 29 and 31, for the quotient may be written (x) • Thus, the differential quotient of "^ is obtained by writing it (l + x^) -' Applying the theorem for products, we get I — x'^ d (l+^2)_ dx '^{i-x^J dx which can readily be reduced. GENERAL THEOREMS OF DIFFERENTIATION 27 If the student prefers, he may simply memorize the result of example 2, § 32, and apply. 35. If 2 is a function of y, and y of ^, an increment £^x of X produces Ay of ^y, which in turn produces A2 of 2. Evidently ^ = ^ . ^. ^x A_y A^ The limits of these magnitudes (assuming that definite limits exist) will therefore have the same relation, viz. : d^_dz^ dy dx dy dx This may also be expressed : If F{x) = ^\_f{x)-\, then F\x) = ^\f(x)\f{x). It must be carefully noted that 0'[/(^)] means the derivative of 0[/(jr)], not with respect to x^ but with respect \.o f{x). It is — not dz ... ^0[/(^)] ^ ^0[/(^)] '^y -y, or agam it is ,\, / not '-^, -* . dx ^ df{x) dx In words, the derivative with respect to x of a function of a function of x, is the derivative of the former function 7m'tk respect to the latter, multiplied by the derivative of the latter with respect to x. Thus, if jj/ = (i + x'^Y^ ^- maybe found by denoting (i + x^) by tv, and then finding -^ from y = w', and — from w= I -\- x'^. Whence t 7 7 dzv dx dx d-w dx ^ ov -r y But the use of w is quite unnecessary, and the student should learn to dispense with it as well as with y also. The required derivative then Employing the notation of differentials, the process is even more easily remembered and applied. The differential o< y^[/(^)] or ^'U{x-)yAx), or ^{Xix')-\l\x\ Substituting for ^ its given value and for ^' its value as obtained by § 35> we have ^w = 0'{(V'[/w])}^'[y(-^)]/w, and so on for any number of functions. If we use differentials instead of differential quotients, we have 40l(02[3(-)])} = <^l'^02 = 0l'02'^03 = etc. The proof is left to the student. GENERAL THEOREMS OF DIFFERENTIATION 29 Examples. 1. Plnd the derivative of 4{2(I + ^2)-2 4- 3(1 + ;,2)3|2 + 5(2(1 + x'^Y + 3(1 + X^Yf. 2. Differentiate {a+[d+{c + Ax")^Yf. 38. The results of this chapter may be thus summarized : dx K4,'(x). '^' dx ~ l{x)Y 30 INFINITESIMAL CALCULUS CHAPTER III DIFFERENTIATION OF THE ELEMENTARY FUNCTIONS 39. We have learned (§§ 16, 33) that the derivative of x"^ is «jc"~^, where n is any integer, x^ is the elementary algebraic function. We have now to differentiate elementary functions called "transcendental." To do this we recur to the general method of differentiation. We first take up the trigono- metric functions. ^0» ^(sin x) _ y sin (^x + A.r) — sin x dx Ax _y sin X cos Ax -f cos x sin Ax — sin x = lim| Ax sin A^ . I — cos Ax cos X sm X • Ax Ax But becomes unity at the limit when Ax becomes zero, and Ax - cos Ax , becomes zero. Ax These are shown by means of Fig. 4, where AB is an arc Ax on a unit radius OA. So that ^C is sin Ajf, CO is cosAjt, and CA is I — cos Ax. !illA£ is therefore ^ Ax BA and i_:^cosA£ .^ CA^ Ax BA DII^FERENTIATION OF FUNCTIONS 31 When BA becomes zero, CA and BC become zero. The proof ,t lim — B. ing hints : that lim = i, and lim = o, is left to the student with the follow BA BA i> f RC CO 1. I > > = , which approaches i as limit arc^^ DA AO ^^ 2 CA^CAB£^B£B£ ^^^^^ approaches o x i. BA BC BA CE BA ^^ Hence o, to discover the nature of F(xi). We shall solve this by reasoning from F'" successively back to F"f F', and F. Since F"'(xi) is positive, it shows that F"(x) is increasing as x increases. But as F"(xi) is zero, the fact that F"(x) is increasing 44 INFINITESIMAL CALCULUS shows that it was negative before reaching F"{xi) and positive after. This is our conclusion for F". Since F"{x) was. negative before reaching F"{xi) it shows that F'(^x) was //len decreasing, and since F"{x) was positive afterward, F'{x) was /Aen increasing. But, if F'(x) is zero at F'(x{) and was decreasing before and in- creasing after, it must have been positive both before and after. This is our conclusion for F'. Since F' is positive both before and after, it shows that F(^x) was increasing both before and after, and is there- fore not a maximum, but a horizontal point of inflection. Thus let F{x) be x*-6x^ + Sx+ 7. Then F' is ^x^ — I2.r + 8. Then F" is I2;»:2- 12. Then F'" is 24 jr. The roots of F' = o are i and — 2. For jr = i, F" vanishes, but F'" is positive. Hence we know that /' or x* — 6^^^ -f 8;f -f 7 is at a stationary inflectional value increasing on either side, as x increases. But for ^ = — 2, F" is positive. Hence for this value of x, F is a. minimum. 59* Examples. — 1, Find maximum or minimum value of .j:^. 2. Find maximum or minimum value of ^x^ — 27 x. 3. Find maximum or minimum value of 2x'^ -{- x -\- i. 4. Find maximum or minimum value of .r^ — 12 jt + 6. 5. Find maximum or minimum value of 2 x^ ■}- 6 x'^ -{- 6 x + ^. 6. Find maximum or minimum value oi x^ — 2x -{- ^x^ — 4. 7. What is the nature o{ x* — 24. x^ -{- 64 x -{■ 10 for x — 2? 8. What is the nature oix^ -{-4x^ + 6 x^-{-4x+ ij ior x =— i ? 60. If F(x) is of the form (f>(x) -f- X, where JC is any constant, then the same values of x render J^(x) a maxi- mum or minimum as render cf)(x) a maximum or minimum respectively. SUCCESSIVE DIFFERENTIATION 45 For the nature of F{x) or of {x) when X is a positive constant, then the values of x which render F{x) a maxi- mum or minimum are the same as those which render <^{x) a maximum or minimum respectively. \{ F{x) = K<^{x) where ^ is a negative constant, then the values of x which render F(x) a maximum or minimum are the same as those which render <^(^) a minimum or maximum respectively. For the successive derivatives of these two functions (viz., K{x) and (t>{x)) are K"{x) and "{x), etc. J [ etc., and evidently the very same values of ;*: will make the two first deriva- tives zero, and, if K be positive, will make the two second derivritives of the same sign or both zero; but if K be negative, will make them of the opposite sign or both zero. Similarly for the two third derivatives, etc. Since the natures of /^and of 0, as respects maxima and minima, depend exclusively on the signs (-f, — , or o) of their derivatives, the theorem is proved. Thus, to obtain the value of x which will make a maximum or minimum, we drop the constant factor (which is evi- dently positive) and find out which values of x make x'^ — x, a. maxi mum or minimum. 46 INFINITESIMAL CALCULUS Examples. — 1. Interpret the theorems of §§ 60, 61 geometrically. 2. Find maximum or minimum of 5(1 -\- x ■\- x^^-\- 10. 3. Find maximum or minimum of — 3;i:(;r+i+ — J. 4. Find maximum or minimum of m | ^(-^ ^bx-\- c')-^ e _,_ ^ | . 62. The subject of maxima and minima is one of the most important in the Calculus, and has innumerable appli- cations in Geometry, Physics, and Economics. Let ABC (Fig. 8) be any triangle, and EFKH a rectangle in- scribed within it. This inscribed rectangle will vary in size according to its position. If too low and flat, it is small. If too high and thin, it is also small. Between these positions there must be a position of maximum, where the area is the largest possible. Now its area is the product of the base HK or EF by the altitude DM, and the problem consists in discovering where EF- DM is a maximum. To do this, we must first express EF and DM in terms of some one variable. Out of the many possible {e,g. BH, BK, AE, EC, EH, HK, etc.) we select AM, and denote it by x. We call AD = ^ and BC=:a. Evidently MD = h — x. To express EF in terms of x, we proceed as follows : The triangles AEF and ABC are similar, so that their bases and altitudes are proportional. That is, AM^EF ^^ x^EF AD BC ""^ h a ' SUCCESSIVE DIFFERENTIATION 47 whence EF= — • h Consequently EF X DM -{h-x) — • h We wish to know for what value of x this expression is a maximum. We may omit the positive constant factor -, leaving h {h — x)x or hx — x^y the first differential of which is h — 2.Xy which, put equal to zero and solved, gives k x = -t 2 the required answer. We are sure it is a maximum and not a minimum or stationary in- flectional value, since the second differential is — 2; i.e. negative. We have learned, therefore, that the maximum rectangle inscribed in a triangle is that whose altitude is half the altitude of the triangle. In physics many important principles depend upon max- ima and minima. Thus the equilibrium of a pool of water, a pendulum, a rocking chair, or a suspension bridge, is deter- mined by the condition that the centre of gravity in each case shall be at the lowest possible point. In economics we have the principle of maximum con- sumer's rent, of maximum profit under a monopoly, etc. 63. Examples. 1. How must a given straight line be divided so that the product of its two parts shall be a maximum ? 2. What is the minimum amount of tin necessary to make a cylin- drical vessel which will have a given capacity A? What must be the relation between the height k and the radius of the base r? 3. Find the maximum cylinder inscribed in a circular cone of revolution. Ans. Altitude of cylinder equals one third that of the 48 INFINITESIMAL CALCULUS 4. Find the maximum rectangle inscribed in a semicircle. Ans. The sides are - \/2, and r^J~2, 2 5. A cylinder of revolution has a given diameter. What altitude must it have in order that it may have the least total area in propor- tion to its volume? Hint. — Express volume and total area in terms of the variable alti- tude Xy and the constant radius r. Then find v\ hen total area . . . IS a minimum. volume 6. If the function pF{p) is continuous, what equation gives a value of/ which makes the function a maximum? Write the algebraic expression denoting the condition under which the value of/, in the equation asked for, corresponds to a maximum or minimum. 7. If the price, /, of an article is fixed and the cost of producing it, for a given individual, is a function IX-^), of the quantity produced, X, how much must he produce to make his profit, xp — F{x), a maxi- mum or minimum? Express this result in words. What condition must L\x) satisfy that the profit may be a maximum and not a mini- mum? Express this condition in words. 8. Four equal squares with side x are removed from the corners of a square piece of cardboard with side c and the sides are turned up so as to form an open square box. If the square box is to be of maxi- £ 6* 9. The distance between two points, B and C, on a coast is 5 miles. A person in a boat is 3 miles distant from B, his nearest shore point. Supposing he can walk 5 miles an hour and can row 4 miles an hour, what distance from C should he land in order to reach C in th**, shortest possible time ? Ans. i mile. 10. Given /, the slant height of a right cone; find the altitude when the volume is a maximum. Ans. - y/^. TAYLOR'S THEOREM 49 CHAPTER V Taylor's theorem 64. We know that certain functions can be developed in terms of powers of variables. Thus {a-\-xy becomes by the binomial theorem a*-\-4 a^x -f- 6 c^x- + 4 ao^ + x^. Again, by simple division, we may show that (provided x lies between — i and + 1) = I — X -\- x^ — x? -\ . I -\- X Now the Calculus supplies a much simpler and more gen- eral method than algebra of developing functions in series of this sort. Thus, let <^(^) be any function of x developable in the form (x)= A 4- B{x - «) + C{x - af + D{x - of + .••, where a. A, B, C, etc., are constants, and the series con- verges. We shall show how to express the " undetermined coefficients " A, B^ C, etc., in terms of the single constant a. By successive differentiation, we have * <^\x) = B ^ 2 C{x - a)^ zD{x - of -\- '" "{x)= +2C -f-2 . 3Z>(^-d;)^-... etc. * By § 26 which can readily be extended so as to apply to an infinite number of terms if, as is here assumed, the sum of these terms con- verges. 50 INFINITESIMAL CALCULUS Since these equations (and the original from which they are derived) are true for any value of x, they are true when They then become (a)=J, or A=(a)', '(a)=i.B, B=ct>'(a); etc., where 2 ! means i • 2 and 3 ! means i • 2 • 3, etc. Substituting these values of A, B, C, D, etc., we have «^ (^) = (^) 4- <^' («) (^ - ^) + <^" («) i^^=;^' 2 ! 65. This series, which is "Taylor's theorem," expresses the magnitude of the function for any value of x in terms of its magnitude and that of its derivatives for any other value of X. Thus if we could write down some exact formula y = (t> (x) for the population {y) of the United States in reference to the time (jr) elapsed since, say 1800, Taylor's Theorem tells us that we could get the population in 1900, (;f), merely from data of the census of 1890. As a first approximation we take the population of 1890 itself, (a). But, as the population has. not remained stationary, we add a correction for the increase within the decade. This increase we first assume to be (^x — a) '{a)y multiplied by the time between the two censuses {x — a). But since the rate of increase (by which is TAYLOR'S THEOREM 51 here meant so many thousand souls per year, not the percentage rate) has not remained stationary, we add another correction ^^'^^ ~ ^J , I • 2 constructed on the supposition that the rate of increase of the rate of increase of population, "(a), known to exist in 1890 has remained constant until 1900. Not content with this, we take into account the rate of increase of the rate of increase of the rate of increase of popu- lation, and so on. 66. Geometrically, the theorem states that the ordinate of any point of the curve y= {x) can be obtained from the ordinate, slope, " curvature," etc., of any other point. Thus, OB (Fig. 9) is x and BD, (x); OA is a and AC, can be ascer- tained purely from the data as to the curve at C, viz. its height, the rate at which this height is increasing (i.e. its slope), the rate at which this slope is increasing (i.e. its "curvature" (§ 53)), the rate at which this " curvature " is increasing, etc., etc. In fact, the theorem states that the ordinate DB is the sum of various magnitudes: first, 0(rt), which is represented by ^5 (for this is the same as AC); secondly, 58' (x — a)\a) = (x — «) (l)'{a)), thirdly, ^^ j ^ k j ^ which is represented by 5'5", when 5" is reached 2 ! by drawing the curve C5", which has the same curvature as the prin- cipal curve CD has at the point C, but retains that "curvature" (with respect to the jr-axis, see § 53) throughout; that is, we approach D by adding successive corrections. 5 is the position Z> would have had if the ordinate of the curve had remained unchanged from C (so that the curve would have followed the horizontal C5) ; 5' is the position D would have had if the rate of increase of the ordinate, i.e. the slope of the curve, had remained unchanged from C (so that the curve would have followed C5') ; 5" is the position D would have taken if the rate of increase of the slope had remained unchanged from C (so that the curve would have followed C8"), etc. 67. If we take the point li instead of C, so that a = o, Taylor's theorem reduces to the simple form {x) = {p) + \o)x + ^ \ + ^ \^ +etc. 2 ! 3 ! This is Maclaurin's Theorem. The student vi'ill observe that (o) is by no means itself zero. It is simply that particular value of (x + A) = cf>(x) + \x)k + <^"(:r)^+ -, 2 ! where x now refers to the abscissa of C instead of that of Z>. TAYLOR'S THEOREM 53 The student will also sometimes see the theorem expressed in the same form, but with y employed in place of h. 69. There are *many applications of Taylor's theorem in economics. Cournot in his Principes Mathhnatiques makes frequent use of it, as does Pareto in his Cours d'economie politique. When /^ is a small quantity, as in some of Cournot's cases of taxation, then the higher powers of h may be neglected, and we have the approximate formula i^i^x ■\-h) = (x)-\-h'(x). This is assuming that if the interval AB is very small, the point 8' will coincide approximately with D. 70. It will be observed that an hiatus was indicated in the demonstration of Taylor's Theorem. This means that it is not always possible to develop {x) in the series proposed, and that the attempt to do so will give a diverg- ing or indeterminate series. It is impossible in so elementary a treatise as this to indi- cate in what cases Taylor's Theorem is applicable. The subject is one of great difficulty, and some of the most im- portant conclusions relating to it have only recently been discovered. 71. To show the application of Taylor's and Maclaurin's theorems, let us use them to develop the function (a + xy, assuming it developable. Since {x) — (ai- x^, \x) = n{a-\-xy-\ %o)=n(n-i)a^-\^ etc. Hence A(x^ - "(oy , 2 ! a result which we already know by the binomial theorem. Again let us develop sin x, assuming it developable. Since 4> (x) = sin ;r (p (o) = o, '(x) = cos;r 0'(o) = I, 0''(;r) = — sin;r "(o) = o, i"(^x) = - cos ;r 0'"(o) = - i. etc. etc. Hence x^ = o + x + o- — -{-'- 3' x^ , x^ x"^ . Again let us take X — a -\- 1 Since (;»:) = , 0(a) = I, ;r — a + I 0'(;c) = - (;r - a + i)-2, 0'(a) = - I, ^"(x) =2{x- a+ i)-\ f'(a) = 2, 0^"(^) = -2 . 3(^ - a + i)-4, 0''''(a) = - 3 !. Hence, by Taylor's Theorem, (x) = I —{x — a) + -^ ^ ^-^^ ^ + •••• 2! 3! TAYLOR'S THEOREM 55 72. Among other important uses of Taylor's and Maclaurin's theo- rems are the evaluations of the fundamental constants e and t. To obtain ^, we develop the function ^. 0(^)=^, 0(0)= I, 0'(^)=^. 0'(o)=i, 0"(^)=^, 0"(o)=i, etc. etc. Since ^(.)=,(o) + *'(o). + *:^ + «f!+.., we have "(x) = -2x + 4x^ - 6x^ + -", 0"(o) = o, iP'"{x)=- 2 + 3 .4^2 _ 5 .6^4 + ..., 0'"(O) = - 2, 0(;^) =2.3.4^^-4.5.6^+ ...., ^^''(O) = O, f>{x) = 2 . 3 . 4 - 3 . 4 • 5 • 6^' + • ••» ^"(o) = + 4 '.. etc. etc. — 2X^ 4' X^ arctan;r = o + x + o -\ ; ho + ^^^^H 3! 5! = ,_f^ + ^_£!+.... 3 5 7 * It is assumed here, without proof, that the proper conditions as t' sec^jr + y^ sec^jt tan jr + — sec2^(l + 3 tan^^) + »- o INTEGRAL CALCULUS 57 CHAPTER VI INTEGRAL CALCULUS 74. We have thus far been occupied with the derivation from F of F\ F'\ etc. But it is possible to reverse this process, and, given F'^\ or any other derivative, to pass back to F^\ F\ F. F\x) was called the derivative of F{x) ; we now name F(x) the primitive oi F'(x). The first process of obtaining F' from F is the subject matter of the differential calculus, of which the preceding chapters have treated. The process of obtaining F from F^ is the subject matter of the integral calculus, 75. In the differential calculus, we saw that the result of differentiation was expressed either in the differential quo- tient F'(x), or in the differential F\x)dx. In the integral calculus it is customary to employ only the latter form. We called F\x)dx the differential of Fix) ; we now call Fix) the integral of F\x)dx. We obtained F'(x)dx from F{x) by differentiation. We obtain Fix) from F\x)dx by inte- gration. The symbol of differentiation was d ; that of in- tegration is j . Knowing that dixr) = 2xdx^ we may write j 2xdx = :^ \ or again, since dFix)=.F\x)dx 58 INFINITESIMAL CALCULUS expresses in the most general manner the process of the differential calculus, ^F\x)dx = F{x) expresses the process of the integral calculus. Both equa- tions state the same fact looked at from opposite directions. The former equation reads, " the differential of F{x) is F\x)dx^^; the latter may be read, "the function-of-which- the-differential-is F\x)dx is F{x)j' for the hyphened words are what is meant by " integral of." The simplest form of the above equation is \ dx = x, »75. The symbol i was originally a long S, which was the old symbol for " sum of" (to-day it is usual to employ the Greek S instead). Integration was looked upon as summation, dy being the limit of Ay, and Ay being a small part of y, the differential dy was conceived of as an infinitesimal part of y. An infinite number of dy's were thought of as making up the y. 77. As d{x^) = ^x^dx, it follows that I ^x^ dx = :x^. But d(x^ + 5) = 3 ^Vjc ; hence I ^x^ dx = x^ -{- $ ; that is, the integral of 3 x^ dx (or the primitive of 3 x^) may be x^ 01 x^ -\- $, and evidently also x^ -{- ij or ^^ + any constant whatever. In general, | F\x)dx is F{x) + C, where C is any arbitrary constant. For the latter expression differenti- ated gives the former (§ 27). An arbitrary constant (usually denoted by C) must there- INTEGRAL CALCULUS 59 fore always be supplied after integrating any differential to obtain the complete integral. 78. There is no general method of integration known corresponding to the general method of differentiation of Chapter I. The only way we arrive at the primitive of a given function is through our previous knowledge of what function differentiated will yield the given function. 79. ^ax^dx = ^^ + C, J n -\- 1 ax^"^^ provided n is not = — i . For the differential of h C n -\- 1 is evidently ax^dx provided « -f i is not zero ; i.e. provided n is not = — i. The rule, therefore, for integrating the simplest algebraic function is to increase the exponent by one, and divide the coefficient by the exponent so increased (and then, of course, to add an arbitrary constant). Thus, (2x^dx is | j«^ + C. 80. Examples. 2x dx = 7 J" (zx^dx=l Ans. I^ + C X Cdx^ J x^ CAdx^ J x^ X dx _ -, 2 x-^dx=l ? Ans. - -i- 4. ^. 60 INFINITESIMAL CALCULUS 8i. It may seem at first that a result involving an arbitrary constant can be of little use. But this is far from true. Though we cannot determine the arbitrary constant from the given differentia], we may have, in any particular problem, information from some other source which will enable us to determine it, and often, as we shall see, we do not need to determine it at all. We may interpret the constant C geometrically by plotting the equation v = Fix) -\- C. To know F\x)dx or F\x) is to know the slope of the curve for any value of x. But evidently the slope of the curve does not determine the curve ; since, if the curve were shoved up or down without change of form, it would have just the same slope for the same value of x. The constant C has to do with the vertical position of the curve. It has nothing to do with its form. 82. We may profitably follow the plan adopted in intro- ducing the differential calculus, and begin by considering a mechanical and a geometrical application. We have seen that, knowing a body falls according to the law j=i6/2, (i) we can show that its velocity at any point is (is . / V Suppose, however, we only know that a body acquired velocity according to law (2), can we pass back to law (i)? As has been said, in the integral calculus it is customary to use the differential form to start with. Accordingly, we write (2) in the form ds = 32 /di. Integrating, we have s =^32 ^d^ = ^ -\-C = 16 /'^ C. (3) INTEGRAL CALCULUS 61 Now, although equation (2) with which we started does not enable us to judge of the value of C, we may evaluate C from outside data. Thus if we know that s is measured from the point at which the body started to fall, we know that when / was zero, s must have been zero too. Putting J- = o and / = o in (3), we have o = o + C, or C = o. After substituting this value of C in (3), the equation takes the definite form J =16 A 83. Of course, C is not always zero. In fact, in the above ex- ample, we might reckon the distance s of the falling body not from the point where it started, but from a point 27 feet above. We then know that when / = o, J = 27. Substituting in (3), we have 27 = o + C or C = 27, and (3) now becomes 5=16/24-27. Evidently the value of C depends solely on what origin we use to measure s from. 84. Similarly, if we know the relation between the slope of a. curve ~ and its abscissa, we can obtain the equation ax of the curve, except for an arbitrary constant which regu- lates the vertical position of the curve. This example is the true inverse of the geometrical illustration in the differential calculus (§ 12). But for the purpose of the integral calculus we prefer another geometrical example. 62 INFINITESIMAL CALCUL US 85. Suppose we have (Fig. 10) a plot oi y=f{pc). Give to X an increment Ajc, viz. AE or BK^ and consider the resulting increment not oi y^ but of the area OABC or z. This increment ^z of the area is evidently the small area ABDE. This small area is the sum of the rectangle ABKE and the tiny triangle BDK. The area of the rec- tangle is the product of its base Ajc by its altitude fipc). So that ^z=/(x)^x-{-BnK. (i) Evidently the smaller we make Ajc, the smaller the area of BDK becomes relatively to the small rectangle, and may finally be neglected, giving the important equation dz —f{x)dx. (2) This is not, of course, a mere approximation. It is abso- lutely exact. INTEGRAL CALCULUS 63 The reasoning just given is to be understood as an elliptical form of the following: Dividing, (i) by A^r, we have £=/^^) + ^- (3) T.. BDK . , ,, Now IS less than rect HK , .^ rect HK Ax ' ^*^' BLT But the area of a rectangle divided by its base is its altitude — in this case DJiT. Hence (3) may be written Az — =/(x)+ something less than DIT. Ax It is evident that when Ax becomes zero, DJ^ becomes zero, and *' something less than DJi^ becomes zero," so that our equation becomes ax which may be written dz — f{x)dx. This equation is often written dz =y dXj or z = \y dXy y being the usual symbol for f{x), the ordinate of a curve. 86. Suppose y or f{x) to be that is, \et y — 2, x^ -\- S ^^ ^^^ equation of a curve. The integral calculus enables us to obtain the area z in terms of the abscissa x. We know that dz = (3 ^^ _|_ ^^ ^^^ z = jc-^ + 5 jc + C. (i) 64 INFINITESIMAL CALCULUS The student may test the correctness of this integral by differentiating it and obtaining (3^^ -f ^dx. It remains to determine C. Since we intended to meas- ure the area z from the j-axis, evidently z vanishes when x vanishes. Putting x and z both equal to zero in (i), we obtain C = o. (If we had measured area from some other vertical than the jj^-axis, the value of C would be different.) Hence (i) becomes z=^ x? -\- ^x. Thus suppose x — y^ then 2 = 42. That is, the area included between the curve y = -^x^ ■\- <^^ the axes of coordinates and a vertical 3 units from the ^-axis is 42 units. If the linear units be inches, the area units are square inches. 87« We see more clearly now than in § 76 why integration was first conceived of as summation. The area 2 is evidently the sum of a great many A2's, and at the limit is conceived of as the sum of an indefinite number of dz^%. The dz is thought of as an elementary strip of area infinitely narrow —the limit of ABDE, 88. The problem of obtaining curvilinear areas was one of the earliest and is one of the most important of the applications of the integral calculus. Previous to the discovery of this branch of mathe- matics only a very few curves, such as the circle and parabola, could be so treated. 89. We are here chiefly interested in the geometrical symboUsm. We have seen that the slope of a curve is the differential quotient of its ordinate (with respect to its abscissa). We now see that the ordinate in turn is the differential quotient of its area (also with respect to the abscissa). For dz=ydx means simply dx INTEGRAL CALCULUS 65 If we wish to make a graphic picture of any function and its derivative, we can represent the function either by the ordinate ^ of a curve or by its area, while its derivative will then be represented by its slope or ordinate respectively. If we are most interested in the function^ we usually employ the former method (in which the ordinate repre- sents the function) ; if in its derivative^ the latter (in which the ordinate represents the derivative). That is, we usually like t J use the ordinate to represent the main variable under consiileration. Jevons in his Theory of Political Economy used the abscissa x to represent commodity, and the area z to repre- sent its total utility, so that its ordinate y represented " marginal utihty " {i.e. the differential quotient of total utiHty with reference to commodity). Auspitz and Lieben, on the other hand, in their Untersuchungen Uber die Theorie des Freises, represent total utility by the ordinate and margi- nal utility by the slope of their curve. 90. The method of integration enables us not only to obtain the particular curvilinear area described, but also an area between two Hmits, as AB and A'B^ (Fig. 10). Evi- dently this area is the difference of two areas OA'B'C and OABC. The first is the value of \ f{x)dx, when OA^ (or JC2) is put for x in the integral when found, while the second is the value of the same integral for x = OA (or x^. This is expressed as follows : and is called an integral between limits, or a definite integral The reason it is called definite is that it contains no arbi- 66 INFINITESIMAL CALCULUS trary constant, for this constant disappears when one of the two integrals concerned is subtracted from the other. Thus, if C/(x)dx be J^(x)-\-C, f{pc)dx s means simply (^^(^2) + <^) - (^(-^i) + C), which reduces to F{x^—F{x^, for C must be taken to be the same in both integrals. The area between the curve 3jr2 + 5, the ;r axis, and the two verti- cals erected at :r = 2 and ;f = 4 is J^'CS-^^ + 5)0'^ = [^3 + 5^ + C]^4_[^ + 5 •*• + C]x=2 = 66, for the C drops out, since for each expression the area is measured from the same vertical, though no matter what vertical. It is usual to abbreviate the expression for limits. Xaf=4 /»4 f(x)dXf we write j f{x)dx. 91^ There are certain general theorems of integration corresponding to the general theorems of differentiation of Chapter II. Of these the two most important are : CKf(x)dx = K C/{x)dx and f[./i(x)±/2(x)± "')yx =f/i(xyx±f/2(x)dx±j*/s(x)dx ±.... The proof of the first is simple, for the integral of the right side of the proposed equation is X(F(x) -{-€), or KF{x)-\-KC or KF{x)-\-C\ where F{x) means the primi- INTEGRAL CALCULUS 67 tive of f{x) and C is an arbitrary constant. But C might as well be written C, since its value is anything we please. The integral on the left is also KFix) + C ; for this differentiated gives Kf{x)dx. The proof of the second is also simple. If we denote the primitives of /i(jc:), /2(^), •••, by Fi(x), ^^(x), •••, it is evident that the integral on the right is F,(x) + C, ± F,{x)-\-C2 ± F,{x)-hCs ± -, or F,(x)±F^x)± '-• -\-C, (i) where C is Ci + C2 + Q, and is therefore arbitrary. The integral on the left is the same quantity (i), for the differ- ential of (i) is (§ 26), d(F,(x) ± F,{x) . . . + C) = dF,{x) ± dFlx) • • • =f\{x)dx ±f2(x)dx ••• =(/i(^) ±fj^x) ••• )dx. 92. Examples. 1. Integrate (i -I- a + b)x^ dx, 2. IntegrSitG X- dx -{- y x^ dx -{■ ^ x^ dx. 3. Integrate {k + 2){cx^ dx + ^x^ dx}. Ans. (h+2)\-x^ + -xT + c\^ '5 7 ' 4. If the velocity of a body increases with the time according to the formula — = 3 /2, find the formula for the distance traversed. 5. How far does it move between the instant when / is 3 seconds and that when / is 5 seconds? 6. Find the expression for the area (corresponding to z in Fig. 10) for the curve whose equation is jj' = 5 ^r-^ + 2. Ans. ^y— + 2x -^r C. t. What is the value of that area for the point where ;r is I? Where ;r is 3? Where j is 22? 8. What is the area between the curve, the jc-axis, and the two verticals erected at ;r = 2 and x = 4? Ans. i(X). 9. Solve the same problems for the c\iT\e jf = x^ + 14; for ^ = x^; fory^ = 4rt';f. 10. Find the area z, for y = a^'; y = log (^ + 5) ; y = sin x. Ans. ^-^ + C; (^+ 5) log(.;t:+ 5) - jr + C; -cosx + C, 68 INFINITESIMAL CALCULUS 93. Just as we may differentiate successively, so we may integrate successively. If we perform the integration I f{x)dx and obtain fxipc), we may then take I f\{pc)dx and obtain y^(jp), and then j f^{x)dx and obtain fj^x^, etc. etc. Instead of writing | f^{x)dx, we may substitute for /^(x) its value | f{x)dxy and we shall have which, however, is usually abbreviated to j j f(x)dx dx, or even to J i /{x)dx^. Similarly, we may write I I \/(x)dx dxdx, or I | l/(x)dx^, etc. We may express the double, triple, etc., definite integrals also. The full form for the double definite integral would be x=o \_yx=h _J which, however, may be condensed to INTEGRAL CALCULUS 69 94. To apply these ideas we recur to our old example of a falling body. Suppose our first knowledge is not j = 16/2 nor — = 32/, but — — = 32; that is, we simply know that the acceleration is a given con- dt'^ stant (32 velos per sec), or to be more general let us call this con- stant^. d[—\ The given equation, — ^ = g, means, as we know, -J^ — L = ^^ or (i) ■gdt, whence, integrating, — =gt-\-C; (l) di but this may be written ds = gt dt -\- Cdtj whence, integrating again, s = \gt'^ -f Ci -\- K, (2) We have still to determine the arbitrary constants C and K. If the distance s is measured from the starting-point, then s and t vanish simultaneously. Substituting zero for them both in (2), we obtain It remains to determine C. To do this we take equation (i) and suppose the body falls, not from rest, but with an initial velocity of u feet per second; then when ds . t IS zero, — is «, dt and (i) then reduces to u = o ■{■ C or C = M. Substituting C = u and A' = o in equation (2), we have s=lgfi + ut, the general equation of falling bodies. 95* The process which we have followed out in detail from the equation dh may be condensed as follows : 70 INFINITESIMAL CALCULUS 90. The simple transcendental integrals are obtained as follows : Since d{€vi\x)= zo'=>xdx^ then \ 0.0% xdx = sinx + C. Since a^(cos jr) = — sin x dx, then \ — sm x dx = co6 ;r + C, whence \ sin (x^dx = — cos;r — C = — cos;«r + C, for C is perfectly arbitrary. Since (x, y) ; and if y is also constant, as 74 INFINITESIMAL CALCULUS Thus, the speed of a sailing vessel is a function of her angle to the wind, if the strength of the wind remain constant. The price of woollen cloth is a function of the price of wool, if the cost of labor, etc., remain constant. 102. Since the terms of an equation can be transposed, it is always possible to gather them all on the left side, thus reducing the right side to zero, y = V^^ + i is the same equation as ^ — .r*^ — i = o. The left member is here a function of x and y. And in general it is evident that any relation between two variables y = F{x) can be reduced to the form <^(x, y) = o. When expressed in the first form, y is called an explicit function of x. In the latter it is an implicit function of x. In like manner, any relation z=F(x, y) can be reduced to the form <^(^, y, z)=o; any relation w = F{x, y, z) to (x, y, z, w) — o, and so on. 103. We have seen that ^{x,y)=o or y=F{pc) can always be represented by a curve with x and y as the two coordinates. So, also, <^{x, y, z)=o or z = F(x, y) can always be represented by a surface with x, y, and z as the t/iree coordinates. Draw three axes at right angles to each other, such as the three edges of a room, meeting at a corner on the floor, the ^-axis being directed, say, easterly, thejj'-axis northerly, and the 2-axis upward. To represent z = x^ -^ 2xy -\- 3 j-, let X have any particular value, such as 2, and y, i. Then 2=2^+2X2Xi-f3Xi"=ii. Find the point in the room which is 2 units east of the corner, i unit north of it, and 11 units above it. This is APPENDIX 75 one point of the required surface. By taking all possible combinations of values of x and y, and finding the result- ing values of s, we can find all points on the surface. 104. When z = F(x, y), we may vary x by Ax, while y remains constant, and thus cause in z an increment denoted by A2. The ultimate ratio of Az to Ax is expressed by bz dF{x,y) dx dx and is called the partial derivative of F{x,y) with respect to JC. Similarly, dz^^dF{x,y) dy oy is the partial derivative with respect to y ; i.e. the derivative obtained by keeping x constant during the differentiation. Observe that the symbol d, denoting partial differentia- tion, is not identical with d. 105. The geometrical interpretation of these partial deriv- atives can be made evident. If on the surface, z=F{x,y), say the surface of a stiff felt hat, we take any given point F and pass through it a vertical east and west plane, the plane and surface intersect in a curve passing through F. The tangential slope of this curve at F (or, as we may call it, the dz E-W slope of the surface itself) is — • For the coordi- nates of F are x, y, z, and those of a neighboring point Q on the curve (and therefore on the surface) are x -f Ax, y, z -f Az, where A^ is the difference between the ^'s of F and Q, and Az the difference between the z's ; the jf's are by hypothesis the same. The slope of the line joining /*and Q is 76 INFINITESIMAL CALCULUS — , and its limiting value, lim — or — , is the slope of A^ A^ ox the curve at P (see § 12); i.e. the E-W slope of the sur- face. Similarly, -^, or — V^-^, is the north and south slope of by ay the surface. These two primary slopes of the surface can be repre- sented by placing two straight wires or knitting needles tangent to the hat at the point P, one in an E-W vertical plane and the other in a N-S vertical plane. If we take any neighboring point R on the surface, its coordinates are x + t^x, y + Aj', z + As, where the A's are the differences of coordinates of P and R. Az Join P and R. Then — represents, not the true slope Ax of the line PR, but its easl and wes/ slope (not, of course, the east and west slope of the surface itself). It is the rate the line ascends in comparison, not with its true horizontal prog- ress, but with its eastward progress. A climber ascending a northeasterly ridge may be rising 5 feet for every 3 of hori- zontal progress, but yet rising 5 feet for every 2 of eastward progress. We have to do with the latter rate, not the former. So also — is the north and south slope of the same line PR. Ay Now let R approach P (along any route whatever upon the surface) until it coincides. The line PR approaches a hmiting position which is a new tangent to the surface (a tangent to that curve in the surface which R traced in ap- Az proaching P). The E-W slope of this tangent is lim -— , called — , and its N-S slope, --- dx dy Representing this tangent by a third wire, we have three APPENDIX 77 tangent wires through P, one in an E-W vertical plane, a second in a N-S vertical plane, and the third, any other tangent. The first has no N-S slope ; its E-W slope is — . The second has no E-W slope ; its N-S slope is — • ^•^ dz dz ^y The third has both kinds of slope, viz., — and — dx dy 1 06. As will be shown, the relation between these various derivatives is dz = -^dx-\--^dy, (i) ox dy which mav be thrown into the forms : dz _ dz . dz dy dx dx dy dx or [ . (2) dz__dz_ dx dz dy dx dy dy . The form (i) has the great advantage of symmetry. It seems, however, to conceal the existence of ^ or — , which dx dy are brought out in (2). These last two magnitudes require merely a word of explanation. -^ is not an upward slope dx at all, as it does not involve the vertical z. It is the incli- nation of the third wire across the floor, the rate at which a moving point on it proceeds north in relation to its east- ward progress. 107. The proof of the formula stated in the last section is as follows : * * In order to master and remember this proof, the student is advised to construct for it some actual physical model. He will then find it extremely simple. 78 INFINITESIMAL CALCULUS We first assume that all wires through P tangent to the surface lie in one and the same plane called the tangent plane. This assumption is analogous to that in § 14, that the progressive and regressive tan- gents coincide. There is an exception if the surface has an edge or vi'rinkle at the given point. Let us take in this plane the three tangent wires above considered, viz. the two primary wires (in vertical planes running E-W and N-S respectively) and the wire obtained as the limiting position of Z*^. Take a point Q' on this third or " general " wire, having coordinates X + A'jf, y + cJy, z + tJz. (The primes serve to distinguish Q^ on the tangent plane from Q on the surface.) Through Q^ pass two vertical planes running E-W and N-S respec- tively. We already have two such planes through P. These four vertical planes cut the tangent plane in a parallelogram, of which PQ is a diagonal and the " primary wires " are the two sides meeting at P. Denote the two vertices as yet unlettered by H and K, the former being in the E-W and the latter in the N-S primary wire. A'z being the difference in level of P and Q^ is the sum of the dif- ference in level of P and H and of H and Q\ just as the difference in level between Mount Blanc and the sea is the sum of the elevation of Lake Lucerne above the sea and of Mount Blanc above the Lake. (It does not matter whether H is or is not intermediate in level between P and Q\ for if not, one of the heights considered becomes negative.) Now the difference in level of P and H is for the difference of level, h, between any two points, as M and N (Fig. 11) is the product of the biope of MN hy the horizontal interval, a, between them (since : slope of MN — - , whence /i = a X slope of a APPENDIX 79 MN^. ^ is known to be the slope of PO , and a';c is the E-W interval between P and Q , and therefore also the E-W interval (or in this case the horizontal interval) between /'and H (since JI B.nd Q are in the same N-S plane). Again the difference in level between ZTand Q is by For — , being the slope oi PK, is also the slope of HQ' parallel to dy PK, and A'y, being the N-S interval between P and Q', is also the N-S (and in this case horizontal) interval between H and Q' (since H and P are in the same E-W plane). Therefore, A'2=-^A';r + ^A'>/. (I)' dx By which is the prototype of the desired result (l). _, . , A's dz dz A'y (2)' This may be written —7— = ^ — h -tt- * tt~' ^ A'x (jx dy A'x Now — - is the E-W slope of the "general tangent" wire PQ'. But we have seen that — is also this slope. Again, —^ is the inclina- dx A'x tion of this same wire across the floor (the rate at which a point moving on the wire proceeds northward relatively to its eastward progress). But so also is -^ (§ io6)- Substituting therefore these dx values for the primed expressions, we have dz _ Sz ■ dz dy^ dx Qx dy dx which may be thrown into the form ^d: dx By dz = ^dx + ^dy. In this, dz is called the total differential of 2, while ^ dx and ^ c^ are \\s partial differentials. It is evident that we should reach the same result if in the preced- ing reasoning we had employed K in the way we did employ H, and 80 INFINITESIMAL CALCULUS vice versa; also that we could have divided (i)' by A'_y instead of by tJx. 1 08. The formula (i) (§ 106), or its two alternative forms (2), enable us to ascertain the direction of any tan- gent line to a surface. Thus, let the surface be = ;ir2 + 2 AT^ + 3 /2, and let it be required to determine any tangent line at the point whose X and y are i and i respectively; z is evidently 6. 1. The primary E-W tangent wire at this point has an E-W slope ^=2;f+2jj/ = 4, found by differentiating the above equation treat- ing y as constant, and has no N-S slope. 2. The primary N-S tangent wire at this point has a N-S slope ^=2x-^6y = S, and has no E-W slope. dy 3. The tangent wire in the vertical plane running northeast and southwest has an E-W slope of (/x dx dy dx = 4 + 8^ ■ dx and a N-S slope of = 4 + 8 X I = 12, dz _Sz _ dx . dz dy dx dy dy = 4 X I + 8 = 12. 4. The tangent wire in the vertical plane running northwest and southeast has the two slopes 4+8(-i) = -4 and 4(-i)+8=+4. 5. The tangent wire in the vertical plane cutting between north and east so as to be advancing north twice as fast as east li.e. so that ^=2), APPENDIX 81 has slopes of ^± = ^^^ .± ax Qx dy dx = 4 + 8 X 2 = 20, and ^ = 5i.^+5f dy ^x dy Qy = 4 X ^ + 8 = 10, and so on for any tangent wire whatever. 109. Examples. 1. Find the slopes of the five sorts above indicated for the same surface at the point for which ^ = 3 and y = 2. 2. At the point where ^ = — i, y =.— i. 3. At the point where x = o, y ■= o. 4. For the surface z = x^-\-x'^-\-x-\-xy+y-\-y^-{-y^ at the point X ^=^0, y = 1, 5. For the surface z = x'^y- 2 xY^ + 3 at the point x = 2, y = '^. 6. On the same surface at the same point, what are the E-W and N-S slopes of the tangent line which progresses northward 3 times as fast as eastward ? 4 times ? 3I times ? 7. Answer the same questions for 2 = log jJ^ + 3* + xy. 110. When we have a function of more than two vari- ables, as w=F{x, y, z), there is no mode of geometrical interpretation corresponding to the curve for y = F{x) and surface for z = F{x, y) (unless, indeed, we posit a " fourth dimension," and speak of a " curved space " of three dimen- sions whose coordinates are x,y^ z,w\). It may be shown, however, in a manner strictly analogous to the process of § 107, but without employing the geomet- rical image, that , bw , , bw , , bw J dw = —— ax H dy -\ dz. ox dy dz 82 INFINITESIMAL CALCULUS This differential equation is elliptical for the three equation^i obtained by dividing through by dx, dy, and dz. The theorem and its proof are extensible to any number of variables. 111. A very important application of the principle of partial derivatives occurs when we have but two varial^les, but y is an implicit function of x ; i.e. when <^{x, y) — o. We are enabled to obtain the derivative --^- without being dx obliged first to transform the implicit function into the explicit form y^^F{x), Thus, if x^ -^ y^ = 25, we may find -^ without changing the equa- tion to the form >' = ± V25 — x^. 112. We know from § 106 (2) that if z = <^{x,y)^ then dz_ ^ d{x,y) d{x,y) ^ dy^^ dx dx dy dx which may also be written in two other forms, as given in § 106. When z is zero, as in the case now being considered, then — is also zero (§ 27, end). Making this substitution in the dx above equation, we obtain dy _ dx dx~ d(Xfyy dy In words : To find the differential quotient of y with re- spect to X when the functional dependence between x and y is expressed in the implicit form y4aT LD2i-A30m-7,'73 General Library (R2275810)476 — A-32 University of California Berkeley