elements of infinitesimal calculus

ELEMENTS

. Infinitesimal Calculus.

by

JOSEPH BAYMA, S.J.,professor of Mathematics in Santa Clara College, S.J., Santa Clara, California.

San Francisco :

A. WALDTEUFEL,737 MARKET STREET.

1889.

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CONTENTS.

Introduction,

PART. ^-DIFFERENTIAL CALCULUS.Section I. Rides of Differentiation.

Algebraic functions of one variable,Transcendental functions of one variable,Functions of two or more variables,Implicit functions,

Section II. Successive Differentials.Maclaurin's formula,Taylor's formula,De Moivre's formulas,Maxima and minima,Exercises on maxima and minima,Values of functions which assume an indeterminate form,

Section III. Investigations about Plane CurvesTangents, normals, etc.,Direction of curvature.Singular points, .Order of contact, osculation,Measure of curvature,Evolutes, ....Envelopes, .....Elements of arcs, surfaces, and volumes,

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CONTEXTS.

Differentials with polar co-ordinates,Spirals, 123

PART II. INTEGRAL CALCULUS.

Section I. Various Mel hods for Finding Integrals.Integration of elementary forms, ,Reduction of differentials to an elementary form,Integration by parts,Integration of rational fractions,Integration of binomial differentials,Integration by successive reduction,Integration of some trinomial differentials,Integration by series,

Integration of trigonometric expressions, .

Integration of logarithmic differentials,

Integration of exponential differentials,

Integration of total differentials of the first order

Integration of the equation Mdx + Ndy = 0,Integration of other differential equations,Integration by elimination of differentials,

Double integrals,

Section II. Application of Integral Calculus to Geometry.Rectification of curves 1"Quadrature of curves, 204

Surfaces of revolution 208

Solids of revolution, 210

Other geometrical problems, 213

Problems solved by double or triple integrals, . . .223

Section III. Application of Integral Calculus to Mechanics.Work 231

Movement uniformly varied, 237

Movement not uniformly varied 238

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CONTENTS. 7

PAGEComposition and decomposition of forces 241Moments, 245Virtual moments, 249Attraction of a sphere on a material point, . . . 251Centre of gravity, 256Moment of inertia 263Curvilinear movement, 272

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INFINITESIMAL CALCULUS.

INTRODUCTION.

1. The Infinitesimal Calculus is exclusively concerned with continuous quantities ; for these aloneadmit of infinitesimal variations. A variablequantity is said to be continuous, when it is ofsuch a nature that it cannot pass from one value toanother without passing through all the intermediate values. All the parts of a continuous quantityare continuous : and, as all continuum is divisible,every part of a continuous quantity, how small soever it be, is still further divisible. In other terms,the division of continuum can have no end.

2. Infinitesimal quantities are sometimes conceived as resulting from an endless division of thefinite. But this is not the real genesis of infinitesimals ; for, in the order of nature, it is the infinitesimal itself that gives origin to the finite. Thus,an infinitesimal instant of duration does not arisefrom any division of time; for it is the instantitself that by its flowing generates time. In likemanner, the infinitesimal length described by amoving point in one instant of time does not originate in any division of length ; for it is the actualinfinitesimal motion of the point itself that by itscontinuation generates a finite length in space.

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10 IXFIXITESIMAL CALCULUS.

Hence infinitesimals of time and of length are notmathematical fictions. They are true objectiverealities. Had they not a real existence in nature,neither the origin nor the variations of continuousmovement would be conceivable.

For the same reason we must admit that continuous action cannot produce acceleration except bycommunicating at every instant of time an infinitesimal degree of velocity : and speaking generally, all continuous quantities develop by infinitesimal moments. Hence the branch of Mathematicswhich investigates the relations between the continuous developments of variable quantities, has received the name of Infinitesimal Calculus, and itsmethod of investigation has been called the infinitesimal method.

This method has been used by the best mathematicians up to recent times. Poisson, in the introduction to his classical Traite de Mecanique (n. 12),says : "In this work I shall exclusively use the infinitesimal method. . . . We are necessarily led tothe conception of infinitesimals when we considerthe successive variations of a magnitude subject tothe law of continuity. Thus time increases by degrees less than any interval that can be assigned,however small it may be. The spaces measured bythe various points of a moving body increase alsoby infinitesimals ; for no point can pass from oneposition to another without traversing all the iiPtermediate positions, and no distance, how smallsoever, can be assigned between two consecutivepositions. Infinitesimals have, then, a real existence: they are not a mere conception of Mathematicians."

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INFINITESIMAL CALCULUS. 11

3. Modern authors often define the infinitesimalas the limit of a decreasing quantity. This definition we cannot approve. For the divisibility ofcontinuum has no limit, and therefore cannot leadto a limit. This is so true, that even those authorsconfess that the limit the absolute zerocan neverbe reached. On the other hand, infinitesimals, inthe order of nature, do not arise from finite quantities: it is, on the contrary, these quantities thatarise from them. The origin of infinitesimals isdynamical ; for they essentially either consist in, ordepend on, motion : and as motion has no otherbeing than its actual becoming or developing, soalso infinitesimals have but the fleeting existence ofthe instant in which they become actual. It is forthis reason that Sir Isaac Newton conceived themas fluxions and nascent quantities ; that is, quantities not yet developed, but in the very act of developing. This is

,

we believe, the true notion ofthe infinitesimal, the only one calculated to satisfya philosophical mind.* So long as it remains truethat a line cannot be drawn except by the motionof a point, so long will it remain true that an infinitesimal line is the fluxion of a point through twoconsecutive positions.

An infinitesimal change may be defined, a changewhich is brought about in an instant of time.Now, the true instant is the link of two consecutiveterms of duration : and it is obvious that betweentwo consecutive terms of duration there is no roomfor any assignable length. Hence the fleeting instant has a duration less than any assignable

* On the modern doctrine and method of limits see the note appended to No. 23.

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12 INFINITESIMAL CALCULUS.

length of duration, that is, it has a duration strictlyinfinitesimal. And in the same manner, every otherinfinitesimal change is a link of two consecutiveterms, or of two consecutive states ; for it takesplace iu an infinitesimal'instant.

4. But here the question arises : How can an infinitesimal quantity be intercepted between twoconsecutive points? -Consecutive points touch oneanother and leave no room between them. Itwould seem, then, that what we call "an infinitesimal" is not a quantity, but a mere nothing. Weanswer that a point in motion has always two consecutive modes of being in space ; for it is alwaysleaving its last position, and always reaching a following position which cannot but be consecutive tothe last abandoned.. Now, it is plain, that if theactual passage from the one to the other were not areal change, the whole movement would be withoutchange ; for the whole movement is but a continuous passage through consecutive points. Butmovement without change is a contradiction. It istherefore necessary to concede that between twoconsecutive points there is room enough for an infinitesimal change, and accordingly for an infinitesimal quantity.

As a further explanation of this truth, let us conceive two material points moving uniformly, theone with a velocity 1, the other with a velocity 2.Their movement being essentially continuous, thereis no single instant in the whole of its duration, inwhich they do not pass from one point to a consecutive point, the one with its velocity 1, the otherwith its velocity 2. But the velocity 2 causes achange twice as great as that due to the velocity 1.

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Therefore the ratio of the two movements is, atevery instant, as 2:1. But two absolute nothingscannot be in the ratio 2 : 1. Therefore the movements comprised between two consecutive pointsare not mere nothings, but are real quantities,though infinitely small. They are, in fact, fluxions,or nascent quantities, or, as the Schoolmen wouldsay, quantities in fieri.

Nor does it matter that these infinitesimals aresometimes represented by the symbol 0. For thissymbol has two meanings in mathematics. Whenit expresses the result of subtraction, as when wehave a a = 0, it certainly means an absolutenothing: but when it expresses the result of division, it is a real quotient, and it always means aquantity less than any assignable quantity : butbecause it has no value in comparison with finitequantities, it is treated as a relative nothing, andis represented by 0. Thus, in the equation

the zero represents an infinitesimal quantity. Thiscan be easily proved. For it is only continuousquantities that admit of being divided in infinitum :and, when so divided, they give rise to none butcontinuous quotients, because every part of continuum is necessarily continuous. Now, the absolute zero cannot be considered continuous.Therefore the absolute zero caji never be the quotient of an endless division. And in this sense, itis true, as the theory of limits affirms, that a decreasing quantity may indefinitely tend to thelimit zero, but can never reach it. On the other

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hand, the above equation gives

a = 0 X oo ;

and this does certainly not mean that the finitequantity a is equal to an infinity of absolute nothings.

5. We have said that infinitesimals have no valueas compared with finite quantities. A few yearsago. an American writer* was bold enough tomaintain that this fundamental principle of infinitesimal analysis is not correct. The principle,however, has been admitted by the greatest mathematicians, and its correctness will not be doubtedby any one who understands the real nature of infinitesimals. The principle, says Poisson (loc. cit.),"consists in this, that two finite quantities whichdo not differ from each other except by an infinitesimal quantity, must be considered as equal ; forbetween them no inequality, how small soever, canbe assigned'''' ; because the infinitesimal is less thanany assignable quantity.

Again, it is plain that the infinitesimal is to thefinite as the finite is to the infinite. Now, the infinite is not modified, as to its value, by the addition ofa finite quantity. Therefore the finite is not modified by the addition of an infinitesimal. That theinfinite is not modified by the addition of a finitequantity, can be assumed as an evident truth : butit can also be demonstrated. Thus, it is shown inTrigonometry that between the angles A, B, C of a

* Mr. Albert Taylor Bledsoe in his Philosophy of Xathematic/t, where he strivesto prore that the infinitesimal method should be abandoned. We arc afraid thatphilosophical readers will not consider his effort a success.

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plane triangle there is the relationtan A + tan B + tan O = tan A tan B tan C /

and this equation, taking A = 90, 5 = 45, C =45, gives

oo -f-2 = co ;which shows that the addition of a finite quantitydoes not modify the value of the infinite.

We may draw from Arithmetic a still plainerproof of our principle. Dividing 1 by 3 we obtain

5 = 0.333333 . . .

and multiplying this by 3, we obtain1 = 0.999999 . . .

In this last equation, if the second member beunderstood to continue without end, the differencebetween the two members will be an infinitesimalfraction viz., unity divided by a divisor infinitelygreat. Now, we can prove, that, notwithstandingthis infinitesimal difference, the equation is rigorously true. For, let the second member of theequation be represented by x; then

x = 0.999999 . . .Multiply both members of this by 10 ; then

10a; = 9.999999 . . . = 9 + x;and from thi?, by reduction, we have

925 = 9, x=l.This clearly shows that the equation 1=0.999999 . . .is rigorously true. It is plain, therefore, that aninfinitesimal difference has no bearing on thevalue of a finite quantity, and that no error is com

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mitted by suppressing an infinitesimal by the sideof a finite quantity.

6. The notions above developed may suffice as afirst introduction to the infinitesimal calculus. Wehave shown

1st. That infinitesimals are not nothings, but objective realities :

2d. That infinitesimals are not limits of decreasing quantities, but fluxions that is

,

quantities inthe act of developing, or more briefly, nascentquantities, whose value is less than any assignablevalue of the same nature :

3d. That infinitesimals may have different relative values, and form different ratios :

4th. That an infinitesimal, whether added to, ortaken from, a finite quantity, cannot modify itsvalue.

As to the different orders of infinitesimals, ofwhich we shall have to speak throughout our treatise, we have here simply to state the fact, that infinitely great, and infinitely small quantities arecapable of degrees, so that there may be infinitesand infinitesimals of different orders, each infiniteof a higher order being infinitely greater than theinfinite of a lower order, and each infinitesimal ofa higher order being infinitely less than the infinitesimal of a lower order. How this can be, onemay not find easy to explain, because both theinfinite and the infinitesimal lie beyond the reach ofhuman comprehension : nevertheless we know, notonly from Algebra and Geometry, but also fromrational philosophy, that such orders of infinitesand of infinitesimals cannot be denied. We knowthat the species ranges infinitely above the indi

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vidual, and the genus infinitely above the species.Substance extends infinitely less than Being, animalinfinitely less than substance, man infinitely lessthan animal. From this it will be seen that thenotion of an infinite infinitely greater than anotherinfinite, is not a dream of our imagination, but awell-founded philosophical conception, familiar toevery student of Logic, and admitted, implicitly atleast, by every rational being.

Let us, then, write the following series :

. . . x , x , x, 1, - , - , - . . . .If we assume x = m, it is plain that the first termwill be infinitely greater than the second, the secondinfinitely greater than the third, and so on. Themiddle term 1 being finite, all the following termsare infinitesimal, and each is infinitely less thanthe one that precedes it. Hence infinites andinfinitesimals are distributed into orders. Thus, ifx be an infinite of the first order, x> will be of thesecond order, Xs of the third, etc. ; and in like

manner - will be an infinitesimal of the first order,x '-5 of the second order, of the third, etc.OCT CO

7. The problems whose solution depends on theinfinitesimal calculus, are generally such that theirconditions cannot be fully expressed in terms offinite quantities. Hence a method had to befound by which to' express such conditions ininfinitesimal terms. The part of the Calculuswhich gives rules for property determining suchinfinitesimals and their relations, is called theDifferential Calculus. As, however, none of such

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infinitesimals must remain in the final solutions,rules were also to be found for passing from theinfinitesimal terms to the finite quantities, of whichthey are the elements ; and to effect this, a secondpart of Calculus was invented under the name ofIntegral Calculus.

Of these two parts of the infinitesimal calculuswe propose to give a substantial outline in the present treatise : and we shall add a sufficient numberof exercises concerning the application of the Calculus to the solution of geometric and mechanicalquestions ; for it is by working on particular examples that the student will be enabled to appreciate and utilize the manifold resources of this branchof Mathematics.

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PART I.

DIFFERENTIAL CALCULUS.

8. Our object in this part of our treatise is tofind, and to interpret, the relations which may existbetween the infinitesimal changes of correlatedquantities varying according to any given law ofcontinuous development. Such a law is mathematically expressed by an equation between thevariable quantities ; and it is, therefore, from somesuch equation that the relative values of the infinitesimal changes must be derived.

An infinitesimal change is usually called a differential, because it is the difference between two consecutive values, or states, of a variable quantity.The process by which differentials are derived fromgiven equations is called differentiation, and theequations themselves, by the same process, are saidto be differentiated. Hence this part of infinitesimal analysis received the name of DifferentialCalculus.

Differentials are expressed by prefixing the letterd before the quantities to be differentiated. Thus,dx = differential of x, d(ax') differential of ax2.

9. When an equation contains only two variables, arbitrary values can be assigned to one ofthem, and the equation will give the corresponding

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values of the other. The one to which arbitrary-values are assigned is called the independent variable, and the other, whose value depends on theyalue assigned to the first, is said to be a functionof the same. Thus in the equation of the parabola,y' = 2px, if we take x as independent, y will be afunction of x.

When an equation contains more than two variables, then all the variables but one can receivearbitrary values, and are, therefore, independent,whilst the remaining one will be a function of allthe others.

Functions are often designated as follows :

V 2 = F(x, y\


SECTION I.

RULES OF DIFFERENTIATION.

10. The function of which we have to find thedifferential, may be either algebraic or transcendental. It is algebraic, if it is formed of expressions obtained by the ordinary operations ofalgebra, as addition, subtraction, multiplication,division, and the formation of powers with constant exponents, entire or fractional. It is transcendental when it contains logarithms, circularfunctions, or exponentials.

Algebraic Functions.

11. Polynomials. An algebraic sum of functions constitutes a polynomial. Let

y = s at -f- bu v + c (1)be a polynomial, in which s, t, u, v are functionsof x, and a, b, c constant quantities. When x acquires an infinitesimal increment dx, the functionss, t, u, v acquire the corresponding infinitesimalincrements ds, dt, du, dv, and y acquires its increment dy. Hence the equation (1) becomes

y-\-dy =s + ds - a(t + dt) + b(u + du)-(v + dv) + c. (2)Subtracting (1) from (2) we shall have

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dy = ds aclt -+- &

INFINITESIMAL CALCUL US. 23

and this substituted in (3) gives

d(uvt) uvdt + utdv + vtdu ;which shows that the differential of the product ofthree functions is obtained by multiplying the differential of each function by the product of theother functions, and by taking the algebraic sumof the results. And the same rule holds when thefunctions are more than three.

13. Quotients. Assume

y=\, (i>s and t being functions of x. When x becomesx + dx, then s, t, and y become s + ds, t-{- dt, andy + dy. Accordingly

dy.

+Subtracting (1) from (2) we obtain

s-\-ds s ids sdtt + dt i~ P + tdt '

But the term tdt has no value by the side of V.Hence we suppress it, and we obtainj jfs\ tdssdt /o\dy, or d(j) = ^ . (3)Therefore, the differential of a quotient of twofunctions of the same variable is equal to the denominator into the differential of the numerator,minus the numerator into the differential of thedenominator, divided by the square of the denominator.

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14. Powers and Roots. Let us have

(1)

s being a function of x, and m a constant. Whenx becomes x + dx, this equation becomes

or, developing by the binomial formula,

y + dy = sm + ffls"-1


d = dVs =ds

2Vs

that is, the differential of the square root of aquantity is equal to the differential of the quantitydivided by twice the radical.

The preceding rules are sufficient for the differentiation of any algebraic function of one variable.

Examples. It is of the utmost importance thatthe student should at once familiarize himself withthe above rules of differentiation, and test, by examples, his practical knowledge of them. Let himwork out the following :

1. y ax' bx -f- ac, dy = (3ax' b) dx,

2. y = (a*+xj-b, dy = (d'+x,)xdx,

3. y = ax?+^, dy=(2ax -^dx,a-\~x , 2adx

A-y=a-=x' dy=&=xj,

5. y = 2ax' Sax\ dy = 6ax (x 1) dx,ax , a(ax)4- ax* ,

6- ^ = (^7' (a-xy dx,a^-bx a(x'-2ax + b)y x a y (x af

xdx8.y=l/a*-x*, dy=--^a,, * (b x) dxy= Va-(b~ ^ * = va>-(b-xy '

.

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._ , (a*-\-2x')dx

_ /a'x' 7, _ 2a'xdx

13. y=(a'-x*) Va'+af, dy =- K ,

14. y = (2aa? - a;')') dy-6(a- x) (2ax - x')* dx,l/x+ Va'+a?

15. y = ^x+ Va'+x\ dy = =-dx,

16. y =x(l+x') VT=x~\ dy = 1+^|g' dx,

/ x v , Sx'dx

18. y=-p^= , ^ = (3-^,a? _


becomess , , ids sdty = p whence ay

On the other hand, the differentials of the auxiliaryfunctions are

, _ xdx _ xdx

and these values, and those of s and t, substitutedin the expression for dy, give the differential interms of x.

Transcendental Functions.

15. Logarithmic Fuxctioits. Let s be a function of x, and let

y = \ogs. (1)When x receives the increment dx, this equationbecomes

y + dy = log (s + ds). (2)Subtracting (1) from (2), we have

dy = log (s + ds)-logs = log *^? = iog (l + *p\

But we have from Algebra

Hence, substituting, and suppressing all the infinitesimals of the second and higher orders, we shallhave

dy, or d. (log s) = M - (3)

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The factor M is the modulus of the system oflogarithms. In the common system, whose base is10, we have M= 0.43429448 ... In the Napieriansystem, whose base is

e = 2.718281828459 . . .

we have M=l. The logarithms used in the Calculus are always Napierian, or hyperbolic, if nowarning be given to the contrary ; and the differential of the logarithm becomes simply

d(log*) = .

16. Exponential Functions. Let a be a constant quantity, and * any function of x. The exponential

y = a' (1)will be readily differentiated by the following process. Take the logarithm of both members of (1),differentiate, and reduce. Thus,

log y - s log a, ^= log a. ds,whence dy = y log a.ds, or

d(a") = log a.a'ds.If a be changed into e, then, since loge = l, we

have simply

d(f) =

'xp(xSo[+\)x%=Hp

,'(,-+)=tip

xpf

'xp^+xSo^=ftp

'xpx^a=Hp

ixx=/i-fi

'xSo\a3=d-gi

\\-x)x9=fl'\\

\xqv(Soj =ftp txqvxpq

X-+-a1 xpvz'^-iSot =ftpX7)

=Hpxpi

ftpxp

xp=fivH.X+XaZA+a+X)%01 ^x+xvzfi

\v+2x/[+x)So\ -dpxp

x1 =Hpxp

Xx~\(xlxSo\x =Hp 'xp(xSo\-f-1)

'(xSo\(3o\ dp txSo\xxp

.a-8ft-L

.

ft,Q

.H-g

.ft-f

ft8.ft'z

93

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30 INFINITESIMAL CALC UL US.

17. Trigonometric Functions. Differential ofsine. Let 5 be a function of x. The equation

when x receives the increment dx, becomes

But cos ds 1, and sin ds = ds. Substituting, andreducing,

Differential of cosine. When y = cos s, we shallfind

dy cos (s + ds) cos s= cos s cos ds sin s sin ds cos s ;

which, because cos ds 1, and sin ds ds, reducesto

Differential of tangent. When y = tan s, weshall have

y sin s, (1)

y-\-dy = sin (s + ds), (2)

and subtracting (1) from (2),

dy = sin (s + ds) sin s= sin s cos ds + cos s sin sin s.

dy = d (sin s) = cos s ds.

d?/ = d (cos s) = sin s ds.

' \cos s/cos s d (sin s) sin s d (cos s)

cos" s

that is,

dy =

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or, reducing,

Differential of cotangefd. When y = cot s, weshall have

, /cos *\ sin * d (cos s) cos s d (sin s)dy = dl -. ) = v-i ,v \sin si sin sthat is

,

, cos' s + sin' s ,dy = ds,sin s

or, reducing,

dy d (cot s) ^-5 a v ' sm s

Differential of secant. If y = sec s = ^-j, weshall have

, . , sin 5 dsdy = d (sec = Differential of cosecant, li y cosec s =

we shall have

cos s dsdy = d (cosec s) - sin s

Differential of versed-sine. If y = vers s

= 1 cos s, then

dy = d (vers s) sin s ds

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Differential of versed- cosine. If y = covers 5= 1 sin then

dy=d (covers s) cos s ds.*EXAMPLES.

. m , mm,1. v = sm x, dy = cos xdx,

2. y = sin' x, dy = 3 sin' x cos xdx,

3. y = sin nx cos nx, dy = n (cos' nx sin' nx) dx,

4. y=\og sin(a x), dy cot (a a;)


18. Inverse Circular Functions. We havehitherto regarded the trigonometric lines as functions of arcs. If we now regard the arc as a function of one of its trigonometric lines, we shall havean inverse circular function. The inverse functionsare designated thus,

sin-1 y, cos~ly, tan-1^, etc.,

and are read respectively, arc whose sine is y, arcwhose cosine is y, arc whose tangent is y, etc. Theequations

s = sin-1 y, s cos^y, s = tan-1 y, etc.,are nothing else than the equations

y = sin s, y cos s, y = tan s, etc.,

presented in a new form. Hence, it is from theselatter that we shall derive the differentials of theformer.

From y = sin s we have found dy cos s ds ;hence

ds = dy _ dycos s Vl-sinV

and, since s = sin-ly, therefore

dismay) = Vl-yfound dyFrom y = cos s we have found dy = sin s ds ;

hence

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34 1NFINITESIMAL CALCULUS.

and, since s = cos-1 y, therefore

d(coa-'y)= ^

dsFrom y tan s we have found dy = ; henceCOS s

ds = di/xos1 s = A , 1 , J 1 + tan' *or


j j sin sds = dy-- :cos s

Vl sina s cosec s Vcoseca s 1 'whence

d (cosec-' y) = y Vy'-iFrom y = vers s we have found dy = sin s

36 ISFIXITESIXAL CALCULUS.

c, , 1 3? , 2dx

2r7 x3. s =8\a-'(2x VT^af), ds.

4. s = cos-1 , ds

V l x7adx

adx5. s = tan-'0)>

0. s vers-'/^Y ds

7. s = Bin- ,- ) as

2dx

9. s = tan-1 a A ~ 00s ^


x and y are made to acquire their respective increments dx and dy, the equation (1) becomes

u + du =f (x + dx,y + dy). (2)Hence, subtracting (1) from (2),

du =/ (x + dx,y + dy) -f (x, y). (3)The meaning of this last equation will be better understood if we add and subtract the term/ (x + dx, y)in its second member, which we then put in theform

du f (x + dx, y) -f (x, y)+/ (x + dx, y + dy) -f (x + dx, y). (4)

It is obvious that the difference /'(x -j- dx, y) fix, y) represents the differential of the function .with regard to x alone ; for this difference arises exclusively from the increment dx given to x. Nor isit less obvious that the difference./' (x -\-dx, y -\- dy) f(x-\-dx,y) represents the differential of thefunction with regard to y alone ; for this differencearises exclusively from the increment dy given toy, the other increment dx being common to bothterms, and showing that the differentiation with regard to x has already been performed.

It follows that the total differential of a functionof two variables must consist of two parts, whichare obtained by differentiating the function firstwith regard to x, considering y as constant, thenwith regard to y, considering x-\- dx (or merely x)as constant. The total differential is representedthus,

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where ^ dx is the partial differential of the function with regard to x, and^dy its partial differential with regard to y. The total differential duis the sum of the partial differentials.

Had we added and subtracted the term f(x, y -\-dy)in the second member of (3), we might have put theequation in the form

du =/(x, y + dy)-f (x, y)+f(x + dx,y + dy) -f (x, y + dy). (6)

Here the difference f(x, y-\- dy)-/ (x, y) represents the differential of the function with regard toy alone, and the other difference f(x-\-dx, y -f- dy) f(xi V + (ty) represents the differential with regard to obtained after the differentiation withregard to y has been performed. Now equations(4) and (6) are identical. The total differential istherefore the same, whether we differentiate thefunction hist with regard to x, then with regardto y, or first with regard to y and then with regardto x. In other terms, the result does not dependon the order of differentiation.

20. We can show in the same manner that thetotal differential of a function of three independentvariables is equal to the sum of the partial differentials obtained by differentiating the function withregard to each of the variables in succession. Let

u =


du = f (x + dx, y + dy, z + dz)


B.u = ^, du = yzdxxzdy-xydzZ 2' '

S.u = x\ogW du = dx\ofr?y + xzdy-ydz.

Implicit Functions.

21. An implicit function is one whose value isonly implicitly given in an unsolved equation.Thus y' 2xy = a' is an implicit function of x;whereas, if we solve the equation, we shall havethe explicit function

y x Va?-\-x'.When the function becomes explicit, its differen

tial is> found by the rules already given ; but, assome equations cannot be readily solved, the function may remain implicit, and its differential isthen to be found by the following process. Let

f(x,?/) = 0

be the given function. Its differential will be

/ (x + dx, y + dy) -f (x, y) = 0,

or, by adding and subtracting the term f (x-\-dx, y),f(x + dx, y) -f(x, y) +f(x + dx,y + dy)

-f(x + dx, y) = 0.

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The tirst of these two differences expresses the differential of the function with respect to x, and thesecond exhibits its differential with respect to y.

TJ*

Denoting the first by - dx, and the second by

-~ dy, we have

dx ' dy "

Hence, the differential of an implicit functionf (x, y) = 0 is obtained by differentiating it firstwith respect to x, as if y were constant, then withrespect to y, as if x were constant, and making thesum of the results = 0.

Thus, from the equation a'y' + 6V a'b' = 0, weshall obtain

-j- dx = 2b'xdx, - dy = 2a'ydy,

and2b'xdx+2a'ydy = 0.

22. By a reasoning analogous to the above itmay be shown that the differential of an implicitfunction of three variables, as

9 fo 2A z) = >will be expressed by the equation

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SECTION II.

SUCCESSIVE DIFFERENTIALS.

23. When a function y=f(x) is differentiated,its increment dy is its first differential. The differential of dy (which is written d.dy, or d'y) is itssecond differential. The differential of d'y (whichis written d.d>y, or d'y) is its third differential ;and so on.

The differential of the independent variable, inasmuch as it is the fixed standard with which thesuccessive increments of the function are compared for determining the rate of development, isalways assumed constant. Hence it does not admitof further differentiation.

If the differential of the function be divided bythe differential of the variable, the quotient will bethe first differential coefficient of the function.*

* The authors who use the metliod of limits conceive the differential coefficientas the limit towards which a certain variable ratio if indefinitely approaching.

Their theory is as follows. Let a certain magnitude y depend for its value onsome variable magnitude x, and suppose the relation between the two magnitudesto be expressed by the equation

y = x'. (1)If x takes the increment h, and if the corresponding value of y be represented by

y\ we shall havey' = (x + W = Js + Hxh + h', (3)

and subtracting (1) from (2),y' -y = 2xh + h',

whence v-^- =2x + h. (3)

"\

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The differential coefficient of the first differentialcoefficient will be the second differential coefficientof the function : the differential coefficient of the

In this last equation, the term 'Zx being independent of A, this increment mayundergo any change of value without affecting 2x. Let, then, h continually decrease

till it becomes 0. The expression for the ratio y y will then be simply 2x.

Hence 2x is the limit toward which the ratio - - approaches as h is diminished :which limit the ratio cannot reach until h becomes zero. Such is the process bywhich differential coefficients are determined in the theory or method of limits.

This theory, though still fashionable in France and elsewhere, labors undergreat radical defects. We remark, first, that when h 0, then also y' y = 0.Hence, at the limit, the first member of the equation (8), though represented by

~- in order to keep a trace of the variables, would really be% Now, to assumedx ' 0that hy dxy and dy are absolute zeros is to assume that the limit has been reached,whereas the theory Itself teaches that h = 0 can never be reached, inasmuch as thehypothesis h = 0 would exclude all idea of change or continuity. If, then, the assumption h = 0 can never be true, how can we assume h = 0, and accept equationsbased on such an assumption?

We must also remember that from ^ = 2b we derive

in which expression, if dx were an absolute zero, what would y be but a mere sumof nothings?

Again, the symbol represents, as is well-known, the trigonometric tangent of

the angle that an clement of the curve makes with the axis of abscissas, the element

itself being represented by (Is = \ dx"1 + dy*. Now, this element is not an absolutezero; for the absolute zero, or a mere point at rest, cannot form any particularangle with the axis. Hence d* must be a real quantity; and, if so, dx and dy arealso real quantities. Accordingly the theory which considers them as limits of decreasing quantities is not consistent with itself.

Prof. Todhunter, a follower of this theory, to eschew objections, declares that hedv

considers the symbol ^ as a whole, and does not assign a separatemeaning to dy

di"/ dx. though he knows that- the student will very possibly (and very reasonably,too) tugpect that some meaning may be given, to dx and dy which will enable him

dyto regard as a fraction. The student, however, might humbly remind the Pro

fessor that the ratio represents only a particular state of the ratio and

that, as this latter, so also the former is a result of division; and, therefore, that dyis a real numerator, and dx a real denominator. And as to the separate meanings ofdy and ete, it is not difficult to see that, if x and y be considered as co-ordinates ofa point in motion, dx and dy will represent the developments which x and y are ac-

d.r

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second differential coefficient is tine third- differential coefficient of the function ; and so on.

For example, let y = an?. The successive differentials of this function will be the following,

quiring in a given infinitesimal instant dt : and if r and v' be the velocities withwhich they develop in that instant, then dx = vdt and dy = v'dt ; whence

dy v'dt _ 4/dx

~vdt

~v

Thus it is plain that dy and dx have their separate meaning, and that the ratio

is the ratio of the velocities with which x and y are developing at a given in

stant. It is evident, therefore, that dy and dx are real and distinct quantities,which correspond to h infinitesimal, and not to the limit h = 0.

Perhaps it will be said that, in the theory of limits, the words " when h = 0 " areonly an abbreviation, for the words " when h is continually diminishing towardszero." This is, indeed, what Professor Todhunter explicitly teaches (Diff. Cole.S 9). But it is obvious that, if h is only diminishing towards zero and never reachesthe limit zero, the theory of limits remains without object, and virtually abdicatesin favor of the old doctrine of infinitesimals. For infinitesimals, as defined by theadvocates of the theory of limits (Duhamel, Diff. Cole.), are just such, variable, magnitudes as tend to the limit zero, though they never reach such a limit.

To this doctrine of limits we must object another serious defect regarding itsmethod of working. The theory needlessly starts by giving to the independentvariable x the finite increment h, thus creating for itself the strange necessity of running for ever after the limit h = 0, which limit the theory declares to be unattainable. This is against nature and against reason. The variable x, when continuouslyincreasing, does not change suddenly into e-|-A, but it changes directly into x-\-dx.What is then the use of travelling the whole distance ft, if it is necessary to travel

it back again? Is it natural, when dx presents itself and is within immediatereach, to wander away from it. with no hope of finding it again, except perhapsafter a long and circuitous journey? Is it not more natural, and therefore morereasonable, to pass from x to x -f- dx directly? The infinitesimal increment dx precedes, in the order of causality, the finite increment h ; hence to derive dx from h,

and to say that dx is the limit of a continually decreasing quantity, is to overturnthe order of causality, just as if we said that the acorn is the limit of a continuallydecreasing oak, or that the most rudimentary human embryo is the limit of a continually decreasing baby.

It will be said that all writers on differential calculus, whatever be their theory,always begin with x-\- h

, and assume A to be a finite quantity. Yes; all authors, inmaking their diagrams, give to h a finite value, because infinitesimals cannot berepresented or shown in a drawing; but, though they mark out a finite incrementin order to make it visible to the student, they do not mean that the increment is

finite: they simply mean that there is an increment, and they declare that what appears as finite in the diagram ihust be considered as a mere infinitesimal, or a

nascent quantity. Hence they have no need of going to and fro in search oflimit unattainable, but they immediately take hold of the nascent quantity dx,

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dp = 3ax'dx, d'y = Qaxdx', d'y = Gadx', d'y = 0,. and the successive differential coefficients will be

dx ' dx* dx' ' dx*

EXAMPLES.

1. y = ax", j = nax"-\ ||[ = n (n -1) ax-\d'vrJL = n(n - 1) (n - 2) ax"~\ . . .

2. y = log(a;+l),g = Fl-1,g=-(Air,)dty _ 1.2 d'y 1.2 3dx'

~

(x + l)a' dx'

~(x 4- 1)4

' *

3. y = sin a?, ^ = cos x, ^ = - sin= cos -5-^. = sin x, . . .

4. y = a" 2 = a* log a'

46 ISFIXITESIMAL CALCULUS.

J= (a; + 3)c-, . . . -Jl=[x + n)


and so on. Now, as x may have any value consistent with the convergency of the series, assumex = 0, and let


hence

= (IH, (H (S)-1.--and substituting in the formula (5),

sin a; -a;x 3.3 + 1.2 3.4.5

~1.2.3.4.5.6.7+

- - -

This series, being differentiated and then dividedby dx, gives

cos = 1 - j-g + - j 2 3 4 5 6 + 2. To develop y = (1 + a;)". We have

y = (l+*), gU^l+aO"-1,g = w fa -1)(^ = n(n-l)(n-2) (l+)-8,

hence

^-w(w-l)(/i-2),\dx>)

and therefore

(l + aOn = . . n(n 1) . . 1) ( 2) . .

3. To develop y = log (1 + x). We have

y = iog(i+a!),g=-iA_,^|/_ _i 1.2r^5-


hence

=.='. (SO =-].(SH----and therefore

CC* >V9 CC^log(l + aO = tf- 2"+g --5 + . 4. To develop y = a". We have

V = a" 2 = a* log a' S= a" (log a)''^ = a*aoga)',

whence

(SO = dog ", .-.

.

and therefore

* = l + (loga)* + ^rf+^rf + . . .If we make a e, whence log a - log e = 1, thenwe have

Taylor's Formula.25. Let w ./(#) and = + Considering

a; and A as two arbitrary parts of a certain line, it is

obvious that, if the line receives an infinitesimal increment, the result will be the same, whether theincrement be attached to the part x or to the part h.In other terms, the result will be the same whetherthe function u'=f(x-\-h) be differentiated with

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50 INFINITESIMAL CALCULUS,

regard to x, considering 7t as constant, or with regard to h, considering x as constant. In the firstcase, the differential coefficient of the function will.

be : in the second case it will be and there-da; dlvfore we shall have

du' _ du' f1 .dx~~ dh' ' ' ' K )

This equation will afford us the means of developing the function u' =f(x-\-7i) into a series ar--ranged according to the ascending powers of Ti,

with coefficients that are functions of x alone.Let us assume a development of the form

u' = P 4- Qh + Rl? 4- Sh* + TV + . . . (2)in which P, Q, R, . . . are functions of x alone.Differentiating (2) with regard to x, and dividing bydx, we shall find

4- h 4- h' 4- h' 4- ft1 4- (3)dx ~ dx' dx ' dx dx dx ' " ' >then, differentiating (2) with regard to h

, and dividing by dh,

|| = Q + 2Rh + SSh'+iTh* + . . . (4)Now, by (1), the first members of (3) and (4) areequal ; hence their second members are also equal,and the coefficients of like powers of h in thosesecond members are equal. Therefore

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But P is the value u of the function when h = 0 ;and therefore dP du; hence we shall have

' _ n _ du d _ 1 d'u _ 1 d*/t.r~U) M-2- dx~*' *-2~d- d&*__ 1 d'u~ '

dx* '' " '

These values substituted in the equation (2) give

+ ^2^4+ ' * (5)This is Taylor's formula. The values attributed tox and h must be such as will- render the series convergent.

EXAMPLES.

1. Let u = x* ; then u' = (x -f- h)n ; and we havedu , d*u . 1N _ ,

g = n(7i-l)(7l-2)a!- . . .hence

= (x + 7i) = + rca*-1 7i +

W ^ ~ 1) xn-* h*

+ n(w-l)(n-2)flf.. y+ . . .

2. 3

2. Let = log a?/ then ' = log (a;

+ 7i); andwe have

du_l d'u _ 1 (Pu_2_ d'u _ 2.3dx

~x ' ate'

.r '

'

(&cs

x' '

a;* '

whence

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/ i / , n i . A A* A* A* .

3. Let u = ex ; then v! = e?+h ; and we have

du , d'u , d"udx~ ' dx'~ '


Scholium. Taylor's formula may be used for the development of afunction u = f(x, y) of two independent variables.

If we begin by giving to x an increment h, we shall have. . du. cPu h? d3u h3fix + h,y) = u + ^h + Wl^ + ^K-3+ . . .

and in this equation, when we give to y an increment k, the lirstmember will become f(x + h, y + k), and in the second member uwill become

du. d'u d3u k3U+dyk + dy~' 2+d^2^ +

'h will becomedx

(du: + ~5r * +

-will become

\dx^~ dy'"~f" dy'

d'u K'

)h,

dx* 2

(d*u dSMli v+ * + dy* 2 + /2"'and so on.

Substituting these values in the above equation, we shall have

du, , dw K' d3u k3f(* + h,S + k) = u+^k+ 5-,- + __+ . . .d/^ dC^)

+ dxh + djT + ~W T + 'd' W VdW Mfc

+

54 1ZFIS1TESIMAL CALCULUS.

In this equation, when x receives the increment h, the first member becomes f(x + h, y + k), as before, and in the second member ubecomes

du . d?u h? d'u h?

^rk becomesdy

d (d*\ d*(du)(du \dy) \dy) h*\dy + dx + dx\ 2

+ )*'

d'u k? .3-5 becomesdy* 2

\dyV . KdyV 7t'

ft + TJi IT +dx " T dx* 2and so on.

Substituting these values in the above equation, we shall have

du, d'u h* d*u 7i3

d'(dU)du , \dy/ , , \dy/ h?kay dx dx* 2

d f--\d'u W KdyV hk*

+ dy* 2 + dx 2+ *

+ . . . (B)

Remembering that u =f(x, y), the equations (A) and (B) may bewritten as follows :

du _ du .

^ '

f(x + h,y + k) -f(x, V) = Jj/k + dxh + -j hk + . . .

/du\., , du du, \dy' , ,f(x + h,y + k)_f{x, y) = Txh + Tyk+ -^-hk + ...

and, as (A) and (B) are identical, we conclude that the two coefficients of hk are identical ; whence

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, (du\\dxJ

d fdu'\Uy>

(0.dy dxWhen h and k are infinitesimal, the equations (A) and (B) reduce

todu , du ,

which is the total differential of the function (No. 19). It is, therefore, the property of such a total differential that the differential

coefficient of ^ taken with respect to y is equal to the differential

coefficient of ^ taken with respect to x.

De Moiwe's Formulas.

26. We have found (No. 24) the three series

cos^ = i_|, + t5|4-2^)T;+ (1)smx=x-^ + ^g-j-g - a3;6fi7 + P)

If we change x into x V 1 in this last series, weshall have

x3 V 1 , #4

(4)

2 2.3 ^2.3.4

1 2.3.4.5

If, on the other hand, we multiply both membersof (2) by V 1, and then add the result to (1), weshall find that the sum

cos x -f- V 1 sin x

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will give a series identical with that of equation (4).Therefore

cos x + V 1 sin x e* v'- '. (5)If we change x into x, we shall have also

cos x V 1 sin a; = e-*v-l. (6)And now let a; = W2, to being any number ; then

cos mz + V^l sin mz = e~v'ri = (e**rr,)m,that is,

cos tos + V 1 sin TO2 = (cos 2+ ^ 1 sin 2)m, (7)as also

cos mz V 1 sin mz = (cos 2 - V7 1 sin 2)m. (8)These are De Moivre's formulas. Each of them

is equivalent to two ; for, after developing thesecond members, the real terms will form an equation among themselves, and the imaginary termswill form another, which being divided by the common factor V 1, will express another real relation. Thus, making m = 3, formula (7) will give

cos 32 = cos' 2 3 cos 2 sin' 2,

sin 32 = 3 cos' z sin z sin' 2,

Maxima and Minima.

37. A function is a maximum when it reaches avalue greater than the values immediately preceding and immediately following, and it is a minimum,when it reaches a value less than the values immediately preceding and immediately following.

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To determine if a function u=f (x) has a maximum or a minimum for a certain value of x, thefollowing plain rules are laid down.

If the function, for a certain value of x, becomesa maximum, it must reach that maximum by a lastincrease, and then begin to diminish. Hence thedifferential du must he positive immediately before,and negative immediately after the maximum.

If the function, for a certain value of x, becomesa minimum, it must reach that minimum by a lastdiminution, and then begin to increase. Hencethe differential du must be negative immediatelybefore, and positive immediately after the minimum.

When du changes from + to or from to +,

so also does the differential coefficient ^- . And,

as a quantity subject to the law of continuity canchange its sign only by becoming zero, or infinity,no value of the variable will give a maximum or aminimum value to the function, unless it reduces

^to zero or to infinity. Hence the roots of theequations

du n -.du

will give all the values of x which can possiblymake u a maximum or a minimum.

Let a be one of such roots. It is yet necessary toascertain whether x = a corresponds to a maximum

or to a minimum. If in the expression of ^ weput first a dx, then a + dx, instead of x, and if

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58 ISFIXITESIMAL CALCULUS.

the first result be positive, and the second negative,it is plain, from the preceding considerations, thatthe value x a will correspond to a maximum ; ifthe first result be negative and the second positive,the value x = a will correspond to a minimum ; butif both results be of the same sign, there will be nomaximum and no minimum for x = a.

Assume, as a first example, the function

which givesg = (a-xf + b,

f=^2(a-x).ax 'Making 2 (a x) = 0, we find x a. Puttinga dx, then a + dx, instead of x, the expression 2 (a x) becomes successively

2 (a (a dx)) = 2dx,- 2 (>i - (a + dx) ) = + 2dx ;

and, since the first result is negative and the secondpositive, we see that x a corresponds to a minimum, which is y = b.

The hypothesis 2 (a x) = x gives xco,which cannot verify the conditions of either a

maximum or a minimum.Assume, as a second example, the function

"

?/ = 2ax x%

from which we have

dy a x= dx ^ V2ax-x'Making (a x) = 0, we have x = a. Putting

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a dx, and then a + dx, instead of x, the expression -f- (a x) becomes

+ a - (a dx) -\-dx, + a (a + dx) = whilst the expression (a x) becomes

(a (a dx) ) = dx, (a (a + dx) ) = + dx.

Hence x = a corresponds to a maximum and to aminimum at the same time, owing to the double

sign of ^ . The maximum is y = a, and the minimum y a.

fl*The hypothesis . = = co gives 2ax x'=0,

whence x 0, or x = 2a. But in both cases y becomes = 0 ; and thus there is no other maximumor minimum.

28. When a function of x can be developed byTaylor's formula, the determination of its maximaand minima can be made to depend entirely on itssuccessive differential coefficients.

Let the function iif (x) be at its maximum orminimum, and let

u' =f (x h) and u" =f (x + h)

be the values of u immediately before and immediately after the maximum or minimum. By Taylor'sformula we have

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whence

u''*(-+ +...)u" U Jl / du . d'n h . dhi KV"dx + dx' 2 + dx* 2.3

If u is a maximum, then u>u', and u>u" ; hencethe two series must be both negative. If u is aminimum, then u

ISFISITESIMAL CALCULUS. 61

term be negative, ua will be a maximum : if it bepositive, ua will be a minimum.

If x a were to give also (^^j =0, then the twoseries would reduce to

u~u- - \a&)a +. ld'u\ h' . /d'u\

2.3.4

and thus they would again have different signs.Hence there could be no maximum and no minimum,

unless =0- In this case the t wo series would

begin by the term (^J;) ^\ which, if positive,would give a minimum, and if negative, a maximum.If this term also were to become = 0, we wouldhave to proceed as before with regard to the subsequent terms of the series. From all this we maydraw the following conclusion :If the first differential coefficient which does not

become = 0 is of an uneven order, the two serieshave opposite signs, and there is no maximum orminimum. If the first differential coefficientwhich does not become = 0 is of an even order, thetwo series have equal signs, and there will be amaximum when their sign is negative, and a minimum when the>r sign is positive.

29. The investigation of maxima and minimamay often be simplified. Thus a constant factor

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62 INFINITESIMAL CALCUL VS.

that affects the whole function can be suppressedin the differentiation ; for, since we have = 0,

the result is independent of any such factor.So also, if we have a function of the form y =

Va*x - bx\ we can make y' = n = a'x bx', whence

INF1NITES1MA L OA LC'UL US. 63

a maximum or to a minimum, the second differential coefficient can he obtained without differentiating the other factors, as in the following example. Let

=PXX*,P, Q, and R being functions of x. The regular differentiation would give

dx * dx 1 dx ' *

Now, if the factor i2, for instance, becomes = 0 fora value x a, it is evident that the differential willreduce simply to

Hence it will suffice, in such a case, to multiply theother factors by the differential coefficient of thefactor which becomes = 0.

Exercises on Maxima and Minima.

30. The application of the preceding principlesto the solution of problems is not difficult, thoughthe student may, at times, experience some difficultyin finding out the mode of expressing the particularfunction which is to be worked upon. A few examples will show how the difficulty may be practically overcome.

I. Required the dimensions of the maximumcylinder that can be inscribed in a given rightcone.

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Let A VB (Fig. 1) be the cone, and suppose acylinder inscribed. Let VC= h, AC=r, VO = y, DO x. The volume V of thecylinder will be expressedby

V=nx'(7>-y).But from the similar triangles A VC and D VO wehave

x : y : : AC : VC ::r : h, andhx

Substituting this value of y in the preceding expression, we have

V= Tx^(r x).Hence

Placing 0, we find the roots # = 0 and x =

^ . The first value placed in the expression ofd'V makes it positive, the second makes it negative. Hence x = 0 corresponds to a minimum, and

2rar = to the maximum required, which will be

V=27

of its base is = -5- .

Its altitude is \ , whilst the radiuso2r

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II. In the line CC (Fig. 2) which joins the centres of two spheres, to find the point from which thegreatest portion of spherical surface is visible.

Let J. be the pointsought for. Drawthe tangents AJ/andAM', the radii CM= r, CM' r', and c' 2 dxdy dy2 2

which may be written as Hollows :

u' ~ =/du , du , \ 1 /dHi , _ d-u 17 d-u ,\~(dih + dyk)+2 ( h + 2 dxdyKk + W *"

) "et0-

/du . du.\ l/ ', and > u", and the twoseries must be negative. If u be a minimum, then u


1 /efu,. d'u ,, d! ,,\

which, if negative, will indicate a maximum, and, if positive, aminimum. Making

the trinomial will take the form

+ 2Bhk + CW,or

JJ2or, by adding and subtracting -j. and factoring,

Now, since h and k are arbitrary, and independent of A and JB, we

cannot assume 7 = 7 ,' and therefore the first term f^ + ^of the factor within the brackets cannot be = 0, and is always posi-

tive. As to the second term, 2 , it is easy to prove that itcannot be negative. For as the sign of (2) must remain unchangedfor all the small values of the arbitrary constants h and k, it followsthat the value of the expression (3) must not pass through zero. But,if we had AC B2 < 0, we might choose for 7t and k such arbitraryvalues as would give

Kk + 2) =m-AC

that is,

(2) would pass through zero. Hence the assumptionAC B2 0, or AC B* = 0; and thus in both cases the factorwithin the brackets will be positive. Hence the sign of (2) will bethe same as that of the other factor A.

It follows that the existence of a maximum or a minimum cannotbe inferred from (2) unless either AC> BP, ox AO = B* ; that is

,

unless

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dx*'

d>f> \dxdy) ' r dxl

'"dy1

~\dxdy) ' ( '

The values of x and y which ought to satisfy either the one or theother of these two conditions must, of course, be taken from theequations (1).

The conditions (3) show that the differential coefficients jjjj and

d^u '

2 must always be either both negative or both positive ; and asaythe sign of (3), or of the trinomial, is always the same as that of

-r-; , it is plain that when is negative the function u will be a

dx* r dx*d?u

maximum, and when is positive the function u will be a inini-dxmum.

But let the student remember that, although the existence of a

maximum or minimum cannot be tested by (3) when the trinomial is

= 0, yet even in this case there may be a maximum or minimum :

but it must then be determined by the sign of the fourth differentialcoefficients, after having ascertained that the third differential coefficients, which have opposite signs, reduce to zero, as the theory (No.38) requires.

Exami>le I. A cistern which is to contain a certain quantity ofwater is to be constructed in the form of a rectangular parallelopipe-don. Determine its form, so that the smallest possible expense shallbe incurred in lining the internal surface.

Solution. Let a3 - its content, x = length, y breadth, anda3

therefore =: depth. The total surface u will boxy

a% a3u = xy + 2 + 3 = a minimum.x y

Differentiating first with regard to x, then with regard to y, we find

du 3a3 du 2a>dx x2 dy y

hence

x'y = 3o3 = y*x, and xy = ay'3 ;

and therefore the base must be a square. The depth will be

a3 a3 - a

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and therefore the depth must be equal to half the length or breadth.Since

d?u _ 4,i* _ 4s . tPu _ iPn _dx*

~ j ~ 2~ ' dy*

~ 'dxfiy

~ '

the first of conditions (3) is satisfied, and u is a minimum.Example II. Find the values of x and y which shall make the

functionit x* + y* boxy* ,

a maximum or a minimum.Solution. Here we find

~ = 4x3 iay- = 0, = Aif 8axy 0 ;

hencex3 = ay'-, y' = Zax, x* = %a?x, x* 2a?

x = ay/2, if - 2a?y/2 = a?y/~8, y = ay/%.And again,

^ = 12x* = 24a5, f"r = 12/ - 8ax = 16a? V2,

and therefore the first condition (3) is satisfied ; and as the sign of

and 's positive, x = a a/2 and y = a//8 make adj;' ay5 vminimum.

In this example, the equations ^ = 0 and ^ = 0 are also satisfied by taking x = 0 and y = 0. With these values of a; and y we

find = 0 and (-, = 0. And since in this case the third differ-dx* dy1

ential coefficients are

d3a . d'u,

and the fourth differential coefficients

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are positive, we conclude that the values x 0 and y = 0 correspondto another minimum.

Remark. When u is a function of three independent variables,the conditions of its maxima and minima are determined by a processanalogous to the preceding, but which is based on the extension ofTaylor's formula to a function of three variables, and is too long tobe inserted here. The result, however, of such an investigation issimple enough. If a function

u=f(x, y, z)has any maximum or minimum, it must give

0 0 _ qdx ~ ' dy~ ' dz ~ 'and the values of x, y, z found from these equations must satisfy thecondition

\ dxi'

dy* \dxdy) \'

\ dx'1'

dz* \dxdz) S

/ d'u dht _ tPu\*\dydz dx* dxdy dxdz)

The function will bo a maximum if the two factors within bracketsin the first member of this inequality are negative; but a minimumif they are positive.

Values of Functions which assume an Indeterminate Form.

31. It sometimes happens that in giving to thevariable a certain value, the function assumes oneof the forms

0 oo ^ 1, 0 X oo , x - oo ,0, w , ^ ~, ~ ~, OXqo .

Thus the fraction - , when x1, takes the formar1

|j , though its real value is 2. The real values of

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functions that assume the form - can be found bythe following process.

Let z and y be functions of x, and let u = be-

come r when x = a. Clearing of fractions, and differ

entiating, we have

udy -f ydu = dz ;

but when x = a, the term ydw disappears. Hence(udy)a = (dz)a, or

M. =().Accordingly, the value of u, when it takes the form

j| for x=a, will be found by differentiating separately the numerator and the denominator of its expression, and substituting in the resulting fractionthe value x = a.

If were again of the form , we would apply again the same process of differentiation, and wewould obtain

and, if necessary, we might continue the same process until a determinate value is reached. Thus

sin cc 0the fraction , becomes when x = 0 ; but by

the process just explained we successively obtain

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Fx sin x~\ Tl cos afl ("sin afl Tcos af| 1

When the function, for a certain value of x,jls-sumes the form |

,

or the form 0 x oo , or the form

^r- . its real value may be found by first reduc-

ing it to the form , and then applying the process

above explained. The reduction is easily obtained

by remembering that go =^

. Sometimes this re-

lo"duction is not needed. Thus the function -- .x awhich takes the form | when x = oo , will give immediately

The form oo oo may also be reduced to the form

jj. For let v and w be two functions of x, whichfor a certain value of x become infinite. Then thefunction u = v w becomes oo oo for that valueof x. But we have

u = v-w = v [l--J= - ;V

and when v = oo and w = co, the function will takethe form

^, provided we have 1

^

= 0. If this

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condition were not fulfilled, then the equation u

= v (l would make the function infinite.Assume

u \ sec x x tan x ;

when x ~, the function takes the form oo .But we may write

and from this we shall obtain (u)n 1.i

32. The indeterminate forms 0, oo, 1", can bereduced by the following method. Let v and w betwo functions of x of such a nature that, whenx = a, they cause the expression u = vw to assumeone of the forms 0, oo , 1. Since v = &ogv (e beingthe base of the Napierian logarithms), we shallhave " = ewlogu. Now, the exponent w log ineach of the three proposed cases takes the form

Ox, which can be reduced to^

, as we have ex

plained.EXAMPLES.

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3. [a;

log a;]0 = 0,

. r X* tan x

~"| _ ;T"

Ll + tan x\l ~ 4 '

5. [(i_aOtanf]=?,

6. [tanglog(2-5)] = .

'

\2x- V5x'-aya~ '

8. [^]0 =1,

tan,-.io. r-V-i =1up cot a;Jo11 \a


SECTION III.

INVESTIGATIONS ABOUT PLANE CURVES.

33. Let y =f(x) be the equation of a plane curveMPL (Fig. 9), and let x = OA and y = PA be theco-ordinates of a point

tesimal increment PQof the curve entails an increment AB of the abscissa x, and an increment QR of the ordinate y.If then we represent by s the portion MP of thecurve, then PQds, whilst AJ3 = dx, and QR= dy. Now, we have

AB = PQ cos &, QR = PQ sin #, PQ' = PR*+ QR',

dx = ds cos j?, dy = ds sin &, ds = \fdx' + dy',and

P of the curve. DrawPT' tangent to thecurve at P, and PX'parallel to the axisOX; and make theangle T'PX' = &.

Fig. 9

Let the point Q beconsecutive to thepoint P. The inflni-

hence

dxds tan &

.

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As the tangent PT is but a secant which meetsthe curve at two consecutive points, it follows thatthe tangent and the curve have a common infinitesimal element PQ, and that the angle which theelement PQ of the curve makes with the axis OX isidentical with the angle # made by the tangent atP with the same axis.

The trigonometric tangent of the angle # is takenas a measure of the slope of the curve at the point

P, and, as tan # = ^ , the differential coefficientof the ordinate of any point of the curve is themeasure of the slope of the curve at that point.

Tangents, Normals, etc.

34. The equation of a straight line passingthrough two given points of a curve, whose co-ordinates are x', y', and x", y", is

v"v'

When the two points are consecutive, as P and Q(Fig. 9), then y" y' = dy', and x" x' = dx'; andthe equation becomes

This is the equation of the tangent to the curve atthe point P, whose co-ordinates are denoted byy\ x'.

Making y = 0, we find for the point T, where thetangent meets the axis OX,

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xX .y ,dx'dp"The subtangent A T is evidently = x' x = y,dx'w

35. The normal being perpendicular to the tangent at the point of contact, its equation can be derived from that of the tangent by substituting

-j. instead of + 7^7

dy' 1 dx' Hence

dx ,

3?y-y'=

is the equation of the normal to the curve at thepoint (x', y').

Making y = 0, we find for the point ilT, where thenormal meets the axis OX,

x --- x'-\-y dx'The subnormal AN is evidently equal to x x'

y dx'

Fig. 10

36. An asymptote to a curve is a line that continually approaches the curve and becomes tangent

to it at an infinite dis-/ / tance. Such a line will,1y ' of course, cut either oneor both the co-ordinateaxes at a finite distancefrom the origin.

Let the straight lineA T (Pig. 10) be an asymptote to the curve LP.Since A T, when infinitelyprolong

elements of infinitesimal calculus

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