V^{^y)^ ^Wj/(-^)] represents a function of three differ- \nent functions of different variables. \n\nP\'unctions of two variables are denoted thus: /(-^j >\'), \nf\\x,y), F{y, z), (p{x, y), tp{x, z), ip^{x, z), etc.; and func- \ntions of three variables by F{x,y, z), ^p{r, s, t), etc. \n\nFunctions of any number of variables are indicated simi- \nlarly by writing all the variables within the parentheses. \n\nIn all cases a functional symbol indicates the same oper- \nations in any one subject. \n\nThus, if f{x, y) = ax -{- by, then f{s, t) = as -\\- bt\\ /{i, 3) \n= 2a -\\- ^b] /(o, m) = bm. \n\nHaving 0(^, z,y) = 2x \xe2\x80\x94 cz-^-y"^, then (p{r, s, t) = \n2r-cs-\\- t\\ \n\nFunctions are frequently represented by single letters; \nthus, \xc2\xb1 y Bi^ \xe2\x80\x94 x^ may be represented by y, giving \n_y = \xc2\xb1 y/ R^ \xe2\x80\x94 x^\\ and/(^, y) by z, giving z = f(x,y). \n\nF{x -\\-y)y f{x + Ji), ^{/ + r^) are forms denoting func- \ntions of the sum of two variables. \xc2\xa77. \n\nILLUSTRATIONS. \n\nHaving /(^) -x^-\\- Px^-\' + Qx^-^ + . . .-\\-U,m which F, Q, \netc., do not contain x, then \n\nf{3bc) = {3bcy^ + F{3bc)^-^ + . . . + ^7. \n\nf{a - x) = {a- xy^ -I- P{a - x)\xc2\xbb\'-^ + . . . -j- i/ \n\nf{o) = o^-\\- Fo\xc2\xbb\'-^ + .:..-{- W \n\n/(.\')=0 + i\'(.^)\xe2\x80\x94 + .. .. + 1/. \n\n\n\n1 8 DIFFERENTIAL CALCULUS. \n\nHaving then \n\n(P{a) = Ao\' + ca,

{x) = sax, and J^(w) = ^^ _ ^j, > \n\nthen ^(^[^(.)]) = \'^^\'\'^^~^-(s\'\'Viy ^^^^^^ \n\n\n\nIf 0(jf, jf) \xe2\x80\x94 2j; + sin_j/, and i}){z) = 2>V !^, then ?^f0(jr, jj/)] = 3y2^-|- sin \nIf /(-^j J\' ^) = 7<^^r^/2, and /"(j)^) = \\^y \n\n\n\nthen \n\n\n\n-^(^[/(-O\'.-)])] = 2a^<\'-V^)\'. \n\n\n\n18. Lines are classed as algebraic or transcendental \naccording as their equations involve algebraic functions \nonly or contain transcendental functions. \n\nAny portion of any line may be considered as generated \nby the continuous motion of a point. The law of its motion \ndetermines the nature and class of the line generated. \n\nLet s represent the length of a varying portion of any \nline in the coordinate plane XY, of which the equation in \nX and y is given. .9 depends upon the coordinates of its \nvariable extremities, and varies with each; but the equation \nof the line establishes a dependence between these coordi- \nnates. Hence, j" is a function of one independent variable \nonly. \n\nIf the line is in space, its two equations establish a de- \n\n\n\nCONSTANTS, VARIABLES AND FUNCTIONS. 19 \n\npendence between the three coordinates of its extremities, \nso that one only is independent. \n\nThe same result will follow if a system of polar coordi- \nnates is used. \n\n19. Convexity and Concavity. \xe2\x80\x94 The side of an arc of \nany curve upon which adjacent tangents, in general, inter- \n\n\n\n\nsect is called the convex, and the other, or that upon which \nadjacent normals intersect, the concave side. A curve, at \nany point, is said to be convex towards the convex side \nand concave in the opposite direction. \n\n20. Graphic Representative of a Function of a Single \nVariable. \xe2\x80\x94 The relation between any function and its vari- \nable may be expressed by the equation formed by placing \nthe function equal to a symbol. Thus, placing /jc equal to \nJ, we have J = fx, which expresses the relations between j \nand X, and therefore between the function fx and its \nvariable x. y \xe2\x80\x94 fx is also the equation of a locus, the \ncoordinates of whose points bear the same relations to each \nother as those existing between the corresponding states of \nthe function and variable. Therefore, by constructing, as \nin Analytic Geometry, any point of this locus, its oj^dinate \nwill represent graphically the state of the function corre- \nsponding to the state of the variable similarly represented \nby its abscissa. The locus thus determined is called the \ngraph of the function. It is important to notice that it is \n\n\n\n20 \n\n\n\nD IFFEREN TIA L CA LCUL US. \n\n\n\n\nthe ordinate of the graph, not the graph itself, that repre- \nsents the function. \n\nTo illustrate, let the line AB be the \ngraph oi fx. Then the ordinate PA is the \ngraphic representative oi fx, corresponding \nto a value of x represented by OF. Sim- \nilarly, /^\'^ .represents /x when x \xe2\x80\x94 OP\' . \nThe ordinates PM and P M\' of the \ngraph MQM\' represent two different \nstates of the function corresponding to \nthe same value of the variable, \xc2\xa7 5. \n\nThe ordinates PM, RN, and SO of \nthe graph MNO represent equal states \nY of the function corresponding \n\nto different values of the vari- \nable, \xc2\xa7 5. \n\nThe graph of a function \nwhich is of the first degree with \nrespect to its variable is a \nright line, otherwise not. \n\nThe graph of a continuous function is a continuous line. \n\n21. Surfaces. \xe2\x80\x94 Any portion of any surface may be con- \nsidered as generated by the continuous motion of a line. \nThe form of the line and the law of its motion determine \nthe nature and class of the surface generated. \n\n22. Let u represent the area of a varying portion of the \nsurface generated by the continuous ^ M \nmotion of the ordinate of any given \nline in the plane XY. \n\nu depends upon the coordinates of \ntlie variable extremities of that portion \nof the mven line which limits it, and varies with each ; but \n\n\n\n\nCONSTANTS, V^ARIABLES AND FUNCTIONS. 21 \n\n\n\n\nthe equation of the given line establishes a dependence be- \ntween these coordinates. Hence, z^ is a function of but one \nindependent variable. \n\n23. Let r=f{v) be the polar equation of any plane \ncurve, as DM, referred to the \npole F, and the right line FS. \nLet u represent the area of a \nvarying portion of the surface, \ngenerated by the radius vector \nrevolving about the pole, u will \nchange with v and r; but v and r \nare mutually dependent. Hence, \nu is 2i function of but one independent variable. \n\n24. Let any line in the plane XY, as AM, revolve about \nthe axis of X. It will generate a sur- \nface of revolution. \n\nThe same surface may be generated \nby the circumference of a circle, whose \ncentre moves along the axis X, with its \nplane perpendicular to it ; and whose radius changes with \nthe abscissa of the centre of the circle, so as to always \nequal the corresponding ordinate of the curve AM. The \nradius of the generating circumference is, therefore, a func- \ntion of the abscissa of its centre. Hence, the generating \ncircumference, and any varying zone of the surface gener- \nated as described, is a function of but one independent \nvariable. \n\n25. The area of any surface with two independent vari- \nable dimensions is a function of two independent variables. \nFor example, the area of any rectangle with variable sides, \nparallel respectively to the coordinate axes X and Y, is a \nfunction of the two independent variables x andj>. \n\n\n\n\n22 DIFFERENTIAL CALCULUS. \n\n26. Having any surface, as ATL, let A BCD = ?/ be a \nportion included between the coordinate planes XZ, YZy \nand the planes DQR and BPS, parallel to them respec- \ntively. Let OP \xe2\x80\x94 X and OQ^^yht independent varia- \n\n\n\n\nbles, u will depend upon x^y, and z\\ but the equation of \nthe surface makes z dependent upon x and j. Hence, u is \na function of but two i7ide pendent variables. Similarly, it \nmay be shown that any varying portion of the surface \nincluded between any four planes, parallel two and two, to \nthe coordinate planes XZ and YZ^ is a function of but two \ni7idependcnt variables. \n\n27. Graphic Representative of a Function of Two \nVariables. \xe2\x80\x94 Placing any function of two variables, as/(x jf), \nequal to z, we have z =/(x, j) which expresses the rela- \ntions between the function and its variables. \n\nThe locus whose equation is z \xe2\x80\x94 f[x,y), is called the \ngrai)hic surface of fi^x^)\'), for the reason that the oj^dinaie \n\n\n\n\xe2\x80\xa2 CONSTANTS, VARIABLES AND FUNCTIONS. 23 \n\nof any of its points will represent graphically the state of \nthe function corresponding to the states of the variables \nsimilarly represented by their respective abscissas. \n\nIt is important to notice that it is the ordinate of the \ngraphic surface that represents the function, and not a \nportion of the surface as in the case described in \xc2\xa7 26. \n\nThe graphic surface of a function which is of the first \ndegree with respect to each of two variables is a plane, \notherwise not. \n\n28. Volumes. \xe2\x80\x94 Any portion of any volume may be con- \nsidered as generated by the continuous motion of a surface. \nThe form of the surface and the law of its motion deter- \nmine the nature and class of the volume. \n\n29. Let any plane surface included between any line in \nthe plane XY^ as AM, and the axis of X be revolved about \n\nX. It will generate a volume of revo- y M \n\nlution. The same volume may be gen- \nerated by the circle, whose centre \nmoves along the axis X, with its plane \nperpendicular to it ; and whose radius \nchanges with the abscissa of the centre of the circle, so as \nto always equal the corresponding ordinate of the curve \nAM. The radius of the generating circle is, therefore, a \nfunction of the abscissa of its centre. Hence, the generat- \ning circle, and any varying segment of the volume generated \nas described, is a function of but one indepe7ident variable. \n\nIt is important to notice in this case, that the generating \nsurface is limited by the ordinates PA and P\' M, corre- \nsponding to the extremities of the limiting curve, which \nordinates are perpendicular to the axis of revolution. \n\n30. Having any volume, as ATL, bounded by a surface \nwiiose equation is given, and the coordinate planes, let \n\n\n\n\n24 \n\n\n\nD IFFERENTIA L CAL CUL US. \n\n\n\nABCD-ON ^=- F be a portion included between the coor- \ndinate planes XZ, YZ, and let the planes DQR and BFS \nbe parallel to them respectively. \n\nLet OP ^ X and OQ=y be independent variables. \nV will depend upon x, y and z ; but the equation of the \nsurface makes z dependent upon x andjj;. Hence, F is a \nfunction of but two independent variables. \n\n\n\n\nIn a similar manner it may be shown that any varying \nportion of the volume included between any four planes, \nparallel two and two, to the coordinate planes XZ and FZ, \nis a function of but two independent variables. \n\n3I\xc2\xbb Any volume with three independent variable dimen- \nsions is a function of three independent variables. For \nexample, the volume of any parallelopipedon with variable \nedges parallel, respectively, to the coordinate axes X, Y \nand Z, is a function of x^y and z ; all of which are inde- \npendent. \n\n\n\nPRINCIPLES OF LIMITS, 2^ \n\n\n\nCHAPTER II. \nPRINCIPLES OF LIMITS. \n\n32. The Limit of a variable * is a fixed finite quantity or \nexpression which the variable, in accordance with a law of \nchange, continually approaches, and from which it may be \nmade to differ by a quantity less numerically than any \nassumed quantity however small. \n\nThus, any constant, as C, is the limit of any variable, as \n/(^), when, under a law, /(;t:) approaches C to within less \nthan any assumed value however small it may be. \n\nVarious symbols are used to indicate a limit under a law. \nThus, assuming that f{x) approaches a limit C as x \napproaches a, we write \n\nlimit /(^) = Lt. f{x) = \\imf{x) = limit /(j*:) = C. \n\nEach form is read, " the limit oi f{pc) as x approaches a."" \n\nAny variable which under a law approaches zero as a \nlimit is called an infinitesimal. Thus, \n\n1^"^^^ [i - cos ^1 = o. \n\nAny variable which under a law can exceed all assumed \nvalues, however great, is called an infinite. It is not a defi- \nnite quantity. \n\n* In this chapter the term variable is used in its general sense (\xc2\xa7 i). \nand includes all functions of variables. \n\n\n\n20 DIFFERENTIAL CALCULUS, \n\nAn infinite cannot be a limit. Thus, \n\nis a form indicating that as x approaches zero, i/x is un- \nlimited. \n\nA tangent to any curve is a limiting position of a secant \nthrough the point of tangency, under the law that one or \nmore of its points of intersection with the curve approach \ncoincidence with the point of tangency. \n\nIn some cases, due to the form of the function or to the \nlaw of change, the variable can never become equal to its \nlimit. Thus, \n\nlimit \n\n\n\nlimit ^+i^ii^_r+_i1 \n\nX "4~ I \nBut % I, for all values of x.^ \n\n\n\nThe circumference of a circle is the limit of the perim- \neter of an inscribed regular polygon as the number of its \nsides is continually increased. The radius is the limit of \nthe apothem, and the circle that of the polygon, under the \nsame law. \n\nAn incommensurable number is the limit of its successive \ncommensurable approximating values. Thus, the terms of \nthe series 1.7, 1.73, 1.732, etc., taken in order, are approach- \ning 1/3 as a limit. \n\nIn all cases, whether a variable becomes equal or not to \nits limit, the important property is that their difference is \nan infinitesimal. \n\nAn infinitesimal is not necessarily a small quantity in any \nsense. Its essence lies in its power of decreasing numeri- \n\n\n\nPRINCIPLES OF LIMITS. 27 \n\ncally ; in other words, in having zero as a limit, and not in \nany small value that it may have. It is frequently defined \nas " ail infiiiitely small quantity "y that is not, however, its \nsignificance as here used. \n\nIn representing infinitesimals by geometric figures they \nshould be drawn of convenient size ; and it is useless to \nstrain the imagination in vain efforts to conceive of the \nappearance of the figure when the infinitesimals decrease \nbeyond our perceptive faculties. Usually one or two auxil- \niary figures representing the magnitudes at one or two of \ntheir states under the law give all the assistance that can \nbe derived from figures. \n\nIn all cases, when referring to the limit of a variable, it is \nnecessary to give the law ; for the limit depends not only \nupon the variable, but also upon the law by which it \nchanges. Under a law, a determinate variable has but one \nlimit ; but it may have different limits under different laws. \n\nAn important consequence of the definition of a limit is \nthat if two variables, in approaching limits under a law, \nhave their corresponding values always equal, their limits \nwill be equal. Thus, for all values of x^ we have \n\n(c^ \xe2\x80\x94 x^^/(a \xe2\x80\x94 x) =^ a -{- Xf \nhence \n\nJ\xe2\x84\xa2\'i (\xc2\xab\'- ^\')/(\'^ - *) = lim [a + x]= za. \n\n33 \xe2\x80\xa2 A variable which^ in approaching a limit ^ ultimately has \nand retains a constant sign cannot have a limit with a contrary \nsign. \n\nFor suppose /(j^) becomes and remains positive, and that \nlimit f{x) \xe2\x80\x94 \xe2\x80\x94 C. From the definition of a limit, f(pc) \nmay be made to differ from \xe2\x80\x94 C by a value numerically \n\n\n\n28 DIFFERENTIAL CALCULUS. \n\nless than C. It would therefore become negative, which is \ncontrary to the hypothesis. In a similar manner, it may be \nshown that a variable always negative cannot have a posi- \ntive limit. \n\n34* If the difference between the cori^esponding values of \nany two variables, approaching limits^ is an infinitesimal^ the \nva: iables have the same limit.^ \n\nLet U and V represent any two variables giving \n\nU- V= S, or U= V+d, \n\nin which S is an infinitesimal. \n\nLet Cbe the limit of C/^, then U =\xe2\x96\xa0 C \xe2\x80\x94 e, in which e is \nan infinitesimal. \n\nSubstituting we have \n\nC:-e=F+d, or C - V = d ^ e, \n\nthe second member of which is an infinitesimal. Hence, C \nis the limit of V. \n\n35. The limit of the sum of any finite number of variables \nis the sum of their limits. \n\nLet U^ \xe2\x80\x94 V, JV, etc., represent any variables, and A, \n\xe2\x80\x94 B, C, etc., their respective limits ; then \n\nl7=A-\xe2\x82\xac, - F= -B+S, IV=C- Go.\'etc, \n\n\'n which e, d, go, etc., are infinitesimals. \n\nAdding the corresponding members we have \n\nC/~ F+lV-{- etc.= A- B -rC-^etc- e + (^- &? + etc. \n\n* In order to avoid the frequent repetition of the expression \'" under \nthe law," it will be assumed, unless otherwise stated, that the chancres \nin all the variables considered together, or in the same discussion, \nare due to one and the same law; that all variables and their functions \nare continuous between all states considered, and that they have \nlimits under the law. \n\n\n\nFEIiVCIFLES OF LIMITS. 2g \n\nHence, \n\nlimit [l/-V+JV-^etc.] ^ A -^ + C+etc. \n\n= \\\\mU \xe2\x80\x94 lim F+ lim W-\\- etc. \n\n36. In general, the limit of the product of any two variables \nis the product of their limits. \n\nLet U and V represent any two variables having A and \nB^ respectively, as limits. \n\nThen U \xe2\x80\x94 A \xe2\x80\x94 e and V= B \xe2\x80\x94 S, in which e and d \nare infinitesimals. Multiplying member by member, we \nhave \n\nC/V=AB-Be-AS-^ed, \nand \n\nlimit [CrV] = AB = limit Cr limit F, \n\nIt follows that, in general, the limit of any power or root \nof any variable is the corresponding power or root of its limit. \n\nThus, limit U" = (limit UY, and limit U~^ \xe2\x80\x94 (limit t^)\xc2\xab, \n\nHaving a"" = N, x and N approach corresponding limits \n\ntogether; hence ^""^-^ = lim JV = lim a"", and lim x = \n\nlog lim JV. Also, since x = log JV, we have lim x = lim \n\nlog JV. Therefore log lim JV = lim log JV. \n\n37. In general, the limit of the quotient of any variables is \nthe quotient of their limits. \n\nWith the same notation as in \xc2\xa7 36, we have \n\nlimit ^ = lim lUV-\'\\ = lim ^7[lim F]- = \'^^^ = |. \n\nWhen B = o, and A ^ o, U/V\'is unlimited. \nWhen ^ = o = ^, the principle fails to determine the \nlimit which by definition is determinate, \n\n38. It follows from \xc2\xa7\xc2\xa7 35, Z^, 37, that, in general, the \n\n\n\n30 \n\n\n\nD IFFEREN TIA L CALC UL US. \n\n\n\nlimit of any function of any variables is the same functio7t of \ntheir respective limits. \nThus, in general, \n\nlimit /(^ V, ) =/(Iim U, lim F, ), \n\nand to obtain the limit of any function of variables we, in \ngeneral, substitute for each variable its limit. \' \n\n39. Exceptions to the above general rule arise, and are \nindicated by the occurrence of some indeterminate form, as \n\n0/0, CO /oO , OCO , 00 \xe2\x80\x94 00 , 0\xc2\xb0, 00 \xc2\xb0, I*. \n\nTo illustrate, having /(:<;) = {pc"\xe2\x80\x94 i)/{x \xe2\x80\x94 i), the general \nrule gives limit /(^) = 0/0, whereas we find \n\n/(2) = 3> /(i-5) = 2.5, /(i.i) = 2.i, /(i.oi) = 2.01, \n/(i.ooi) = 2.001, etc., \n\nand the nearer we take x to i, the nearer will /(:^) approach \nto 2. By taking jc sufficiently near to i,/(^) may be made \nto differ from 2 by a number less numerically than any as- \nsumed number however small. Hence, 2 is (\xc2\xa7 2^2) the limit \noi fi^x) as x-Wf-^Y. It should also be ^^bserved that 2, con- \nsidered with the states of /(^) which immediately precede \nand follow it, conforms to the law of continuity. \n\n\n\nY \n\n\n\n\n^ \n\n\nM\' \n\n\n\n\n\n\nw \n\n\n^ \n\n\n^;=^=^ \n\n\nA?/ \n\n\n\n\n\n\n,y \n\n\n\n\n\n\n\n\n/b/ \n\n\n/ \n\n\ny \n\n\n\n\ny \n\n\n\n\n^^ \n\n\n\n\np \n\n\nAX \n\n\np\' \n\n\nX \n\n\n\nH \n\nTo illustrate a failing case geometrically, let the curve \nBMM\' be the graph of a function. Take any state, as \n\n\n\nPRINCIPLES OF LIMITS, 3 1 \n\nPJ^ corresponding to ^ = OP, and increase x by PP\' rep- \nresented by L.X. Draw the ordinate P\' M\' and the secant \nMM\'. Through M draw M Q\' parallel to X. Q\'M\', de- \nnoted by Ay, will represent the increment of the function \ncorresponding to Ajt:. \n\nQ\'M\'/PP\' = Ay/ Ax = tan Q\'MM\' will be the ratio \nof the increment of the function to the corresponding in- \ncrement of the variable. \n\nAt J/ draw MX tangent to the curve. Then, under the \nlaw that Ax approaches zero, the secant MM\' will ap- \nproach coincidence with the tangent MT, and the angle \nQ\'MM\' will approach the angle Q\'MT, or its equal XHT, \nas a limit. \n\nHence \n\nlimit {Ay/ Ax) \xe2\x80\x94 lim. tan Q\'MM\' = tan XHT, \n\nwhereas the general rule gives o/o as a result. \n\nWe observe from the above illustration that tAe limit of \nthe ratio of any increment of any function of a single variable \nto the corresponding increment of the variable, wider the law \nthat the increinent of the variable approaches zero, is equal to \nthe tafigent of the angle made with the axis of abscissas by a \ntangenty to the graph of the function, at the point correspond- \ning to the state considered. \n\nWhen J/\' coincides with J/ the secant may have any one \nof an infinite number of positions other than that of the \ntangent line MT, for the only condition then imposed is \nthat it shall pass through M. \n\nTherefore, while limit (Aj/Ajc) is definite, and equal to \n\nthe tangent of the angle that the tangent line at J/ makes \n\n\n\n32 DIFFERENTIAL CALCULUS, \n\nwith X, limit Aj/limit A^ = o/o indicates that the tangent \nof the angle which the secant makes with X becomes inde- \nterminate when M\' coincides with M. \n\nLimit (Aj/Aj^) is, therefore, one of the many values that \nlimit AJ^^/limit l\\x may have under the law. \n\nThe exceptional cases, in general, require transformation \nin order that factors common to the numerator and de- \nnominator may be cancelled, or from which the limit may \notherwise be determined. They are of the highest impor- \ntance, for the Differential Calculus, as it will be seen, is \nbased upon the limit of the ratio of the increment of the \nfunction to the corresponding increment of the variable \nunder the law that the increment of the variable vanishes. \nThe remainder of this chapter will, therefore, be devoted to \ncertain important exceptional cases and methods. \n\n40. ^^^^^ :sr--j^ ^ my\xe2\x80\x94 ^ \n\nx\'m-^y X \xe2\x80\x94 y \n\nThis formula is deduced in Algebra for all commensu- \nrable values of m. Since (\xc2\xa7 32) any incommensurable \nnumber is the limit of its successive commensurable ap- \nproximating values, the formula holds true when m is \nincommensurable. \n\nLimit ^_Q^ a^ j aP^_a_ \n\nAs X \xc2\xbb^^ CO , it reaches a value k > a, thereafter \n\n\n\n< 1; also 1 \xe2\x80\x94 \n\ni \\x \n\n\n\n< \n\n\n\n\xc2\xab" la\\ \n\nkW \n\n\n\nbut \n\n\n\nlimit ^ll\xc2\xb1Y~"^Q . limit ^ ^ q. \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n33 \n\n\n\n42. Limit(i + yy\'y \xe2\x80\x94 e. \n\n\n\n2 \n\n(i-;;)(i-2j;). . . [i-(^-iM ^^^ \n\nAs j^\' B-^ o each term approaches, as a limit, the corre- \nsponding term of the series \n\ni + i + ^4-,-^ + ...+ l^ + etc., \n\nthe sum of which is shown in Algebra to be ^ = 2.71828 . . . \n43. Limit a^ \xe2\x80\x94 I \n\n\n\nh-^-^o \n\n\n\nlog a. \n\n\n\nPlace a!" = 1 + J, whence h = loga(i + ^), and j/b-^/^m^o ; \ngiving \n\n\n\nlimit \n\n\n\nlimit \n\n\n\ny \n\n\n\nh^o h ~-^^\xc2\xb0log.(i+j) \n\nT \n\n= lim \n\n\n\nlog^. \n\n\n\nlog.(i + yY\'y \\ogae \n44. If unity is the limit of the ratio of any two -variables, \n\nthe limit of any function of one will be equal to the limit of \n\nthe same function of the other. \n\nLet C/" and ^represent any two variables, giving X\\v^\\iU/V \n\n= I. Then \n\n\n\nlimit/(C^) = lim/[^]=/( \n\n\n\nlim \n= /(lim F) = lim/(r). \n\n\n\nV \n\n\n\nlim V\\ \n\n\n\n34 DIFFERENTIAL CALCULUS, \n\nThus, \n\nlim[C:+ VX = lim [C -f ^] ; lim {CU^, = lim [CV} \n\nlim C ^ = lim C^ ; lim [ U/ W] = lim [ V/ IV]. \n\nTherefore, in searchi?ig for the limit of any function under \na law, we may replace any variable entering it by another vari- \nable, provided that, under the satJie law, unity is the limit of \nthe ratio of the two variables interchanged. The great ad- \nvantage in so doing arises when it enables us to determine \nthe required limit more readily. Thus, in the last example \nabove we may be able to determine the limit of F/^more \nreadily than that of U/W. \n\nIn making the above substitution it is important to notice \nthat the limits only are equal, and that corresponding values \nof the quantities interchanged, in general, are not equal to \neach other. \n\nThe privilege of replacing one variable by another under \nthe conditions described, so facilitates the determination \nof limits in certain exceptional cases, that it is important to \ndetermine under what circumstances the limit of the ratio \nof two variables is equal to unity. \n\n45* In general when lim \xc2\xa3/"\xe2\x80\x94 lim F, \n\nlim lU/VX^Xim U/Xim V= i. \xc2\xa737. \n\nThat is, in general, unity is the limit of the ratio of any \ntwo variables when, under the same law, they have the same \nlimit, or, what is equivalent, when the difference between \ntheir corresponding values is an infinitesimal. \n\nIf, however, lim 17=^ lim F = o, it does not follow that \nlim [CI/ V] = I {\xc2\xa7 39). Such cases require special investi- \ngation, and the following are selected on account of their \nsubsequent impoitance. \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n35 \n\n\n\n\\ \n\n\nM \n\n\nQ\' \n\n\nQ \n\n\nV \n\n\nm\' \n\n\nP \n\n\nAX \n\n\nP\' ^ \n\n\n\nQ ^ \n\n\n\' \n\n\n; \n\n\nL.^^^"^ \n\n\nP\' X \n\n\nP Aa^ \n\n\n\n46. Take any plane surface, as PMM\'P\\ included be- \ntween any arc, as MM\', the ordinates of its extremities, \nand the axis of X, \n\nThrough J/ and J/\', Y \nrespectively, draw MQ\' \nand M\'Q parallel to \nX, and complete the \nrectangle MQM\'Q\'. \nLet y = PM, and y\' = P\' M\' . \n\nThen as PP\'= Ax, b-> o, we ultimately have \n\nPQM\'P\'\'%PMM\'P%PMQ\'P\\ \n\nand y\'M-^y. \n\nTherefore ^^ IML^l = n^ IA^ = , \nand limit \\PMM\' P\' / PMQ\' P\'\\= i. \n\nHence (\xc2\xa7 44) \n\nlimit PMM\'P \' ^ j.^ PMQ^P \' ^ ^^^y_Ax _ \nAx:^->o A^ Ax Ax \n\nIf the coordinate axes make an angle with each other, \nthen \n\n\n\n=y- \n\n\n\nlimit \n\n\n\nPMM\'P\' ,. ysin Bax . ,. \n\n= lim =y sm c/. \n\n\n\nAx \n\n\n\nAx \n\n\n\n47\xc2\xab Let MPM\' be the surface generated by the radius \nvector PM = r, revolving about \nP, as a pole, from any assumed \nposition, as PM, to any other, as \nPM\'. Let Av represent the \ncorresponding angle MPM\' . \nWith /^ as a centre, and the radii \nPM and P M\' , describe the arcs \nMQ\' and M\' R respectively. \n\n\n\n\n36 DIFFERENTIAL CALCULUS. \n\n\xe2\x80\xa2Then, as Az^ ^W)-^ o, we ultimately have, in any case, \narea RPM\' ^ area MPM\' % area MPQ% \nand limit [area RPM\'Jzx^d. MPQ\'] = i. \n\nHence, I\'^^o [area MPM\'/sirGSi MPQ\'] = i. \nTherefore (\xc2\xa7 44) \n\n\n\nlimit z= lim ^^ = hm \xe2\x80\x94 \n\n\n\nA57B^-^0 \n\n\n\nl\\V \n\n\n\n/\\V \n\n\n\nAv/2 __ r^ \n/\\v 2 \' \n\n\n\n48. Let FMM\' P\' be any plane figure as described in \xc2\xa746, \n\nand MQM\'Q\' the cor- \nB responding rectangle. \n\nQ\' Revolve the entire \n\nM^ figure about X. \n\n\xe2\x80\x9e/ Then as Aj\\;b->o, we \n\nr \xe2\x80\x94 ^x \n\n\n\nM \n\n\n\nP Ace \n\n\n\n.M \n\n\n\nPax \n\n\n\nultimately have \n\nVol. gen. by > Vol oren. by > Vol. gen. by \nPQM\'P\' < PMM\'P\' < PMQ\'P\\ \n\n\n\nBut \n\n\n\nlimit [ Vol. gen by /VoL gen. by~| \nAx:^o L PQM\'P\' I PMQ\'P\' J \n\n\n\nlim \n\n\n\nny^ Ax __ \nTZy^ Ax \n\n\n\nHence, \n\n\n\nlimit FVol. gen. by /Vol. gen. by"1_ \n\\x^^o L PMM\'P\' I PMQ\'P\' _\\- ^\' \n\n\n\nTherefore (\xc2\xa7 44) \n\n\n\n.. . /Vol. gen. by\\ /Vol. gen \\ \n\nlimit / pMM\'P\' . I PMQ\'P\' ) \n\n\\ AX / ^ Ax ^ \n\n\n\nlim \\ny^ Ax/ Ax\\ = Tty^, \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n37 \n\n\n\n49. Let MJVM\'N\' be a portion of any surface included \nbetween the coordinate planes ZX\\ ZY and the two planes \nN\'SE and NP D parallel to them respectively. \n\nDenote the corresponding volume MNM\'N\\ OFFS \nby V. Construct the parallelopipedons OFFS-OM and \n\n\n\n\nOFFS-FM\', and represent their volumes by F and F\' \nrespectively. Let OF = h, OS = k, OF :=^ /, OM = z, \nand FM\' = z\' . \n\nThen as h^m^k-w^ o, or what is equivalent, as /\xc2\xbb^-\xc2\xbb o, \nwhence 2\' ^^2r, V will, in any case, ultimately be, and \nremain, between F and F\' . \n\n\n\nBut \n\n\n\nHence, \n\n\n\nlimit .p/pr \nlimit \n\n\n\nlimit \n\n\n\n\\.VlF-\\ \n\n\n\nIzhk/z\'hk] = I. \nand (\xc2\xa7 44) \n\n\n\nJ^o [^Aect. (9/^7^^] = lim [/^/M] = liiv \'zhk/hk\\ = z. \n\n\n\n^8 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n50. Let MNM\'N\', denoted by S, be a portion of any \nsurface included between the coordinate planes ZX, ZY \nand the two planes N\' SE and NPD parallel to them \nrespectively. \n\nLet OP = h, and OS = k. \n\nAt M draw the tangents MB and MB\' to the curves \nMN and MN\' respectively, complete the parallelogram \nMBQB\', and denote it by T. T is the portion of the \ntangent plane to the surface at M included between the \n\n\n\n\nplanes which limit S. Let /5 equal the angle which T \nmakes with XY, giving T cos (3 = OPFS. Inscribed \nin S^ conceive an auxiliary surface, composed of 11 plane \ntriangles the sum of which, as ;/ ^-^ co , will have 6" as a \nlimit and such that the sum of their projections upon XY \nwill equal OPFS, The two triangles MN\' M\' and MNM\' \nin the figure illustrate a set fulfilling the conditions. Let /, \n/\', etc., represent the areas of the triangles and ^, ^\', etc., \nthe angles which their planes respectively make with XY. \n\n\n\nPRINCIPLES OF LIMITS, 39 \n\nThen OPFS = ^/ cos ^ = r cos /?, \n\nand :Et cos 0/T cos P = 1 (i) \n\nAs ;z ^-> 00 , S remaining constant, each triangle is an \ninfinitesimal, and we have \n\nThe same effect and result follows if -5" is made infini- \ntesimal and n remains constant. Hence, under the law \nthat A and k vanish, or, what is equivalent, that OJ^y repre- \nsented by /, is infinitesimal, we have \n\n^[s/2^] = ^ W \n\nUnder the same law, /3 is the common limit of 6, 6\\ etc. \nHence (i) \n\njl^\'^ l^t cos e/T COS /?] = lim [^f/T] = i. . (3) \n\nTherefore (2), (3), ^^\\[S/ T] = i, and (\xc2\xa7 44) \n\n^ Ji\'^jLo [^M^]= li\xc2\xab^ i^M^I = lim [{M/cos (5)/hk\\ \n\n\xe2\x80\x94 I /cos ft. \n\n51 \xe2\x80\xa2 Unity is the limit of the ratio of an angle to its si?ie, of \nan angle to its tangent^ and of the tangent of an angle to its \nsine, as the angle approaches zero. \n\nLet OCM = given in radians \nbe any angle less than 71/2 ; then \ntan > > sin 0, and as ;^-> o, \nwe have always C \n\ntan ^ \n\nsm sin \n\nB\xe2\x80\x9ej limit ta,L0 ^ ,i\xe2\x80\x9e^ -J- = I. \n\n<*s^o sm cos

0/tan > sin 0/tan 0, we have \n\n52. U\'m\'fy IS the limit of the ratio of any arc of a7iy curve \nto its chords as the arc approaches zero. \n\nLet s denote the length of any arc of any curve, and \nconceive it to be divided into n equal parts, the consecutive \npoints of division being joined by chords forming an in- \nscribed broken line whose length is designated by/. \n\nThen }:t\\^IP\\ = ^. \n\nUnder the above law the equal arcs of s vary inversely \nwith n\\ hence the same effect and result may be caused by \nretaining any fixed value for \xc2\xab, and making s approach \nzero. Hence, \n\nand if \xc2\xab= I, J^^^Jarc/chord] = i. \n\n53. Unity is the limit of the ratio of the surface generated \nby any arc of any curve under a law to that generated by its \nchord as the arc approaches zero. \n\nLet s^p and n denote, respectively, the same quantities \nas in the last article, and conceive s and;> to move together, \nunder a law, so as to generate two surfaces represented by \nS and P respectively. \n\nAs ^/\xc2\xab^^oo, any state of s without regard to form is the \nlimit of the corresponding state of/, and 6" is the limit oi P. \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n41 \n\n\n\nThat is, \n\n\n\nlimit \n\n\n\nls/p-\\ = I. \n\n\n\nAs ;2B-^oo, the equal arcs of s approach zero; hence the \nsame effect and result may be caused by retaining any fixed \nvalue for 7i and making ^:^^o. Therefore \n\n\n\nlimit \n\n\n\nVs/P^ \n\n\n\n\\in \n\n\n\nlimit \n\n\n\nSur. gen by an arc \n\n\n\n\'^^^ LSur. gen. by its ch \n\n\n\nre ~j \n\n\n\nThe results determined in \xc2\xa7\xc2\xa7 51, 52 and 53, with the \nprinciple \xc2\xa7 44, are of great value in finding the limits in the \nfollowing exceptional cases. \n\n54. \\.^\\MM\'~s be any arc of any plane curve, T^il/ \nand F\'M\' the ordinates of its extremities. \n\n\n\n\n\n\n\n\n\n1 \n\n\nr \n\n\n\n\nY \n\n\n,>C \n\n\nL\' \n\n\n\n\n\n\nM \n\n\n^ \n\n\n^^=^^ \n\n\n., \n\n\n\n\n\n\nY \n\n\n\n\n\n\n\n\n^^/ \n\n\nV \n\n\nV \n\n\n\n\n\'n\' \n\n\n\n\ny^ \n\n\n\n\np \n\n\nAx \n\n\np\' \n\n\nX \n\n\n\nH \n\nThrough J/" draw the chord MM\' = c, the tangent MT \n= 3, and MQ\'= FP\'\xe2\x80\x94 Ax, parallel to X. \n\n1-1 .^.^/^ 1 ^ s\'mMM\'T \n\nFrom the triangle MM\' T, we have \xe2\x80\x94 = -: \xe2\x80\x94 TT-r^TTi\' \n\n^ \' ^ sm M7M\' \n\nAs Ajic approaches zero, the arc s and the angle M\'MT \n\nvanish, but the angle T remains constant. Hence, the \n\nangle J/J/T approaches [180"\xe2\x80\x94 T], and \n\n\'b \n\n\n\nlimit \n\n\n\n^ limit fsin^M^I ^ sin_( \nj\xc2\xab^o |_ sin T J ! \n\n\n\n80\xc2\xb0- r) \n-\xe2\x96\xa0 \xe2\x80\x94 ^= \xe2\x80\x94 ^ \xe2\x80\x94 I. \n\nsm 1 \n\n\n\n42 \n\n\n\nD IFFEREN 1 \'I A L CALC UL US. \n\n\n\nHence, since (\xc2\xa7 52) \n\nlr.\':^A] = i, we have (\xc2\xa744) iH [V^] - l \nTherefore (\xc2\xa7 44) \n\nb ,. FaVccs Q\'MT\'^ I \n\n\n\niri-ii=\xc2\xab-^=i.\xc2\xbb.L- \n\n\n\n] \n\n\n\nt^x J COS Q\'MT \n\n55* Let r ^ f{v) be the polar equation of any plane \ncurve, as AMM\\ referred to the ri^ht line PD, and \npole P. \n\n\n\n\nLet AM = ^ be any portion of the curve, and PM = r \nthe radius vector corresponding to M. \n\nRegarding ^ as a function of v (\xc2\xa7 18) let v be increased \nby MPM\'\xe2\x80\x94ixv. The arc MM\' will be the correspond- \ning increment of s. Draw MQ\' perpendicular to PM\\ \nand denote PM\' by r\' . Then (\xc2\xa7\xc2\xa7 44, 52) we have \n\n\n\nlimit arc MM \' _ ch. M M\' \n\n\n\nAv \n\n\n\nAv \n\n\n\nlim \n\n\n\n/mQ\'" 4- Q\'jW \n\n\n\n_ . i /(^ sin A e\')"^ +(/\xe2\x80\x94;- cos AvY \n\n\n\nPRINCIPLES OF LIMITS. \n\n\n\n43 \n\n\n\nHence (\xc2\xa7 51) \n\n\n\n\n\n\nWe also have \n\n\n\ntan FMD = }\'\'^\'^ tan Q\'M\'M=\\im ^-^, \n\n\n\nt\\Vl \n\n\n\n= lim -7 = lim \xe2\x80\x94. \n\nr \xe2\x80\x94 r cos Av r \xe2\x80\x94 t \n\n\n\nHence, \n\n\n\nlimit r - r \n\n\n\nwhich, substituted in (i), gives \n\n\n\nlimit arc M M\' ^ ^ ^, _^ \n\n\n\nAz\'B^O AZ^ \n\n\n\ntan\' FMB \n\n\n\nIf the radius vector PM coincides with the normal to \nthe curve at M, we have \n\ntan FMI? = 00 , and }^\'^^ [arc MM\'/ Av] = r. \n\n56. Let any plane figure, as j \n\nPMM\'P\\ included between any \nplane arc, as MM\'\\ the ordi- ^ \nnates of its extremities and the \naxis of X, be revolved about X. \nThen (\xc2\xa7\xc2\xa7 44, 53) \nlimit Sur. gen. by arc MM\' / \n\n= lim ^^""^ ^^\'^\' ^^ \'\'^\' ^^\' ^ lim "^^-^ ^^\'^\'^ \nAx Ax \n\n__ \xe2\x80\xa2 \'^{y + y ) ^ Vcos QMM* _ 2 ny \n\n"" A^^ cos (2\'JO^* \n\n\n\n\n44 DIFFERENTIAL CALCULUS. \n\n\n\nCHAPTER III. \n\nRATE OF CHANGE OF A FUNCTION. \n\n57* A function changes uniformly with respect to a vari- \nable when from each state all increments of the variable \nare directly proportional to the corresponding increments \nof the function. \n\nIt follows that from all states equal increments of the \nfunction correspond to equal increments of the variable; \nalso that the ratio of any increment of the function to the \ncorresponding increment of the variable is constant. \n\nThus, having 2ax^ increase x by any amount denoted by \nh, then 2a{x -\\- K) \xe2\x80\x94 2ax \xe2\x80\x94 2ah will be the corresponding \nincrement of 2ax, It varies directly with h, it is the same \nfor all states of the function, and 2ah/h = 2^ is a constant. \nHence, 2ax changes uniformly with respect to x. \n\n\'Let fx be any uniformly varying function, and /i any in- \ncrement of X. /{x -\\- /i) \xe2\x80\x94 fx will be the corresponding \nincrement of the function, and \n\n[/(^ + ^) \xe2\x80\x94 fx\\/h = constant = A. \n\nHence /(x -\\-A)=A/i-{- f{x), \n\nin which .r = o, gives f{h) = Ah -\\- /(o). \n\nTherefore, fx = Ax +/(o). \n\n\n\nRATE OF CHANGE OF A FUNCTION. \n\n\n\n45 \n\n\n\nHence, all functions which change uniformly with respect \nto a variable are algebraic, and of the first degree with \nrespect to that variable, and all algebraic functions of the \nfirst degree with respect to a variable change uniformly \nwith that variable. \n\nThe graphs of such functions are right lines, and any \nfunction whose graph is a right line changes uniformly \nwith respect to the variable. \n\nTo illustrate, the right line AD is the graph of a function. \nConsider any state, as that represented by the ordinate PA, \n\n\n\n\n\n\n\nc \n\n\nc \n\n\n\n\nB \n\n\n/ \n\n\nQ" \ns" \n\nP" \n\n\nA \n\n\ny^ \n\n\nQ\' \n\n\n\n\n^ \n\n\nP \n\n\np\' \n\n\n?\'" X \n\n\n\nIncrease the corresponding value of the variable, repre- \nsented by OP, by any increments, as PP\' and PP" . Q\' B \nand S"C will represent the corresponding increments of \nthe function, and the similar triangles AQ\' B and AS"C \ngive \n\nAQ; : AS\'\' :: Q\'B : S"C. \n\n\n\nThat is, the corresponding increments of the variable and \nfunction are proportional. \n\nBy giving to jc = OP any equal increments, as PP\' , \nP\' P" ,P" P"\\ in succession, the corresponding increments \n\n\n\n4^ DIFFERENTIAL CALCULUS. \n\nof the function, Q\' B, Q" C, and Q\'\'D, are equal to each \nother. \n\nIt is also evident that the ratio of any increment of the \nfunction to the corresponding increment of the variable, as \nQ\'BjPP\', or S"C/FF\'\\ or \'q"\'DIP"F\'\'\\ is constant. \n\n58. Having the function 2.v, \xe2\x80\x94 \xe2\x96\xa0 \n\n\n\n"=\' dv dv dy \n\nBut \xe2\x80\x94 = \xe2\x80\x94 = I, hence -- =r - = \xe2\x80\x94 . \n\ndx d/i ax dh ds \n\n\n\n^2 DIFFERENTIAL CALCULUS. \n\n\n\nCHAPTER V. \nDIFFERENTIATION OF FUNCTIONS. \n\nFUNCTIONS OF A SINGLE VARIABLE. \n\n78. The differential of any function of a single variable \nmay be determined by applying the general rule, \xc2\xa7 70, \xc2\xa7 62, \nand multiplying the result by the differential of the variable; \nbut by applying the general rule, \xc2\xa7 70, \xc2\xa7 62, to a general \nrepresentative of any particular kind of function, there will \nresult a particular form, or rule, for differentiating such \nfunctions, which is generally used in practice. \n\n79. The differential of any power of any function with a \nconstant exponent is equal to the product of the exponent of the \npower, the function with its exponent diminished by unity, and \nthe differential of the function. \n\nLet J/ = x^, in which x is any variable and n is any \nconstant. Then, increasing x by h, we have j[\xc2\xa7 70) \n\n\n\ndy __ limit \n\n\n\nHx + hY - ^" "I \n\n\n\nPlacing X -{- h = s, whence h = s \xe2\x80\x94 x, and as ^ b-> o, \ns B-> X, we have (\xc2\xa7 40) \n\n^ ^ limit p"-^n = ^^n-1 ^^^ dx" = nx"-idx. \n\nHaving y\'^, in which y is any function of any variable, \nas xy, we have (\xc2\xa7 77) \n\ndy^dx = {dy^\'/dy) X (dy/dx). \n\n\n\nDIFFERENTIA HON OF FUNCTIONS. 73 \n\nHence, \n\ndy-^/dx = nf-^(dyldx), and tT ^ ny^-^dy. (i) \nSubstituting \\/n for \xc2\xab in (i), we have \n\ndyV\xc2\xab = -y/\'*-!^ = ^-y\'^dy -= dy/n |/y^^. \n\nHence, //^^ differential of the n^^ root of any function is \nequal to the differential of the function divided by n times the \nn^^ root of the n \xe2\x80\x94 1 power of the function. \n\nEXAMPLES. \n\n1. ^ V^ = dx/2. VJ. 12. a\' f ^ = dx/l ^7K \n\n2. dx^ = 2xdx. 13. dx^^ = \xe2\x80\x94nx-^-^dx. \n\n3. dx^ = 3x^dx. 14. \' =f(x), we write (\xc2\xa777) \n\n\n\nDIFFERENTIA TION OF FUNCTIONS. 77 \n\nd\\ogy __ dXogy^ ^ \xe2\x80\x94 L x -^ \n\ndx . dy dx y dx \n\nHence, d log y = dy/y. \n\n\n\nEXAMPLES. \n\nI. flTlog ^ = {de^)/e\'\'. 2. ^ log sin jf = (aT sin x)/sm x, \n\n3. d log x\'^ = dx\'^/x^ = 2xdx/x^ = 2dx/x. \n\n4. :^;<:/(;c2 \xe2\x80\x94 I). \n\n7. ariog[(i +^)/(i - x)] =4(1 + xj/{i - x)]/[(l + ^)/(l - AT)] \n\n=z2dx/{i \xe2\x80\x94 x"^). \n\n8. o h h ^ \n\nHence, da^ \xe2\x80\x94 a^ log adx. \n\n\n\nDIFFERENTIATION OF FUNCTIONS. Jg \n\nHaving a^, in which j- = /(x), we have (\xc2\xa7 77) \n\nda^ _ day ^V _ y ^ ^y \n\ndx dy dx dx \' \n\nTherefore da^ = a^ log ady. \n\nIt follows that d^y = eydy. \n\nEXAMPLES. \n\n1. da^"^ = a^^ logadx^ = 2a*\' Jt: logadx, \n\n2. c/J""^ ^ = a^\xc2\xb0^ ^ log a ^ log X = a^\'^S ^ log a dx/x, \n\n3. da^^ = a"^ \\oga d ^\'x \xe2\x80\x94 a ^^ log a dx/2 Vx, \n\n4. da ^ \xe2\x80\x94 a ^ \\ogad{l/x) = \xe2\x80\x94 a ^ [ogadx/x"^. \n\n5. da^\'x\'^ = a^dx-^ + x\'^da^ \xe2\x80\x94 a^x\'^-\\x log a + \xc2\xab)^. \n\n1 _ 1 \n\n6. de~ "" = e ^dx/x"". \n\n7.^ a\'[(^^^ \xe2\x80\x94 e-x)/2\\ = (^* + e-^)dx/2. \n\nX _X X X \n\n8. ^[a(^" + ^ \xc2\xab)/2] = (.\'\xc2\xab - e\'"^)dx/l, \n\n9. ^i?* log X = (i/x -f- log x)e^dx. \n\n10. \xc2\xab\'[(^* - \\)/{e^- + I)] = le^\'dxl^e^ + i)^ - \n\n11. a\' log \\[e^ - i)/{e^ 4- i)] = 2e\'\'dx/(e^^ \xe2\x80\x94 l), \n\n12. ^" log (^* 4- ^-a-) = (^a^ _ e-\'\')dx/(e\'\'-]-e-^). \n\n13. a\' log {e^ \xe2\x80\x94 e-^) = {e^ + e-^)dx/{e^ - e-% \n\n14. 4(a* \xe2\x80\x94 i)/(a* 4- i)] = 2a* log a a\'x/(o* -1- l)\xc2\xab. \n\n15. a\'(a* + xf = 2(a* + x)(aa; log a -\\- \\)dx. \n\n16. a^^*(i \xe2\x80\x94 A^) = ^*(i \xe2\x80\x94 3^2 \xe2\x80\x94 x3y^_ \n\n17. flt[(^ - e-^)/{e^ + ^-^0] = 4^V(^* + ^"^)*- \n\n18. a[x/{e=\' - i)] = [^^(i ~ x) - \\\\dx/{e^ - 1)2. \n\n19. dx\'^{\\ + xY \xe2\x80\x94 nx\'^-\\\\ + x)"-i(i + \xe2\x96\xa0zx)dx. \n\n\n\n8o DIFFERENTIAL CALCULUS. \n\n20. When jr \xe2\x80\x94 o, find the inclination of the curve jv = lo\'" to X. \n\nAns. 66\xc2\xb0 31\' 30". \n\n82. Logarithmic Differentiation. \xe2\x80\x94 The differentiation \nof an exponential function, or one involving a product or \nquotient, is frequently simplified by first taking the Na- \npierian logarithm of the function. \n\nThus, let u = f^xvi which y and z are functions of the \nsame variable. \n\n.\xe2\x80\xa2. log u=^ zXogy \n\nand (\xc2\xa7 80, \xc2\xa7 75) \n\ndu/u = zdy/y -f- log ydz. \nHence, du = dy\'\' = zy^-^dy -{- y"" log ydz^ \n\nwhich is the sum of the differentials obtained by applvinq \nfirst the rule in \xc2\xa7 79, then that in \xc2\xa7 81. \n\nEXAMPLES. \n\n\n\ni/{x 1)\' log \xc2\xab = I log (x - I) \n\n\n\nV{x - 2f V{x - 3)\' \n\n- I log (^ - 2) - I log {x - 3), \ndu ^ dx 3

x)=*\'2 \xe2\x80\x94 _^2/y. \n\n19. /(^) = x\'/{i - x). /iv (^) = 24/(1 - xf. \n\n20. y = e^ cos ^. d^y/dx\'^ = 2^1\'^e^ cos (jc -|- nit/^. \n\n21. J = \xc2\xb1 j/i?^ \xe2\x80\x94 ^2. ^ \xe2\x80\x94 \xe2\x80\x94 \n\n\n\ndx\'\' \xc2\xb1 {R-\' - x-\'f^\'\' \xc2\xb1 y \n\n22. f{x) = tan X -f- sec :r. /"{x) = cos x(i \xe2\x80\x94 sin x)-^. \n\n23. jf/ =;>r^. dy/dx^ = ^^(i + log xf + X* -1. \n\n24. /(x) = x^ log X. /i^ (x) =r 6x-i. \n\n25. ^ = sin-y d^x/dy* = {qy + 6/)/(i -y)7/2 . \n\n26. jK = log sin X. Dxy = 2 cos ar/sin^ ;\xc2\xbb;. \n\n\n\n27. JJ/ = |/sec 2;f, d^/dx^ = 3(sec 2x)^\'^ \xe2\x80\x94 (sec 2xyi^. \n\n28. j)/ = {x^-\\-a\'\')ian-\\x/a). d^y/dx^ \xe2\x80\x94 ^a^/{a? + x^- \n\n29. _j/ = sec X. d\'^y/dx\'^ = 2 sec^ x \xe2\x80\x94 sec x, \n\nd^y/dx^ = sec Jf tan jr(6 sees x \xe2\x80\x94 l). \n\n\n\n30. \n\n\n;^ = \n\n\n^Jl-lJog X. \n\n\ndy/dx\xc2\xbb \n\n\n\' = \\n - \\/x \n\n\n\xe2\x80\xa2\xe2\x80\xa2 \n\n\n31. \n\n\n/(^) \n\n\n= (2ax)^\'b. \n\n\n/"(2) = \n\n\n4(a-&)/^(^ _ \n\n\n^)a(a+&)/&/^2. \n\n\n32. \n\n\n>\' = \n\n\ntan-i (i/x). \n\n\n.*. X = \n\n\ncoiy. \n\n\n\n\n\n\ndy _ \ndx \' \n\n\n,. I \n\n\n- sin^y, \n\n\nI \n\n\n\n\n\n\n-i + x\'^ - \n\n\n\xe2\x80\xa2\xe2\x96\xa0 (1+^ \n\n\n2)n/8 - Sin"j/. \n\n\n\n\ndiy \ndx-" \n\n\nd sin^ y \n\n~ dx \n\n\n2 siny cosy dy \n\n\n= sin 2y sin^jv. \n\n\n\n\n\n\ndx \n\n\n\nd^y ^(sin 2j)/ sin^ j) sin 2jj/ 2 sin y cos ;j/ <^^ 4" ^i"^ y cos 2;j/ 2c/i/ \n^jf*^ dx dx \n\n= 2(sin jF)(sin 2y cos j -|- cos 2y sin }^{dy/dx) \n=r \xe2\x80\x94 2 sin^_y sin 3^, \n\n\n\n12/ \n\n\n\nS UCCESSI VE D IFFEREN TIA TION. \n\nSimilarly, d^yldx^ = |3 sin* j sin 4^, \n\nand d^yldx^ = (\xe2\x80\x941)\'\' \\n \xe2\x80\x94 i sin\'^/ sin ny. \n\nSince tan-^ x = 7t/2 - tan"\' (i/x), \n\nwe have ^"(tan-^ x)/dx\'^ = (- i)\xc2\xab-\'|W\xe2\x80\x94 \xc2\xa3 sin\xc2\xbb^ y sin ^zy, \n\n1 n/2 \n\nor ^\xc2\xbb^(tan-J ;*:)/^x\xc2\xbb^ = (- jY-i \\n - i sin (w tan-ii)/(i +;f2) . \n\n\n\n34. J = aV(\xc2\xab\' + \xe2\x80\xa2^\')\' \n\n35. _y= aV2 + C"/+C". \n\n36. ;/=^ + ^(x-af/^ \n^j//^;p = 3^/5(x - \xc2\xab)2/5. \n\n\n\n^7/^;c = a + C", of V^;\xc2\xab^"^ = C- \nd\'^y/dx\'^ = - t>c/2SKx - dfl^. \n\n\n\n37. fx=x\'^\xc2\xb1 X^/\\ f\'x = 2X\xc2\xb1^X^\'\'\'l2, f\'x = 2\xc2\xb1 15x1/74. \n\n38. fx = e\'/\\ f\'x = - //7x2, f\'x = (2x + i)e\'^yx\'\' \n\n\n\n39. fx =g \n\n\n\n\xe2\x96\xa0l/x \n\n\n\n/\'x=e-\'/yx\'. f\'x = e-\'\'\\Y - 2x)/x\\ \n\n\n\n40. The relation between the time, denoted by /, and the distance, \nrepresented by s, through which a body, starting from rest, falls in \na vacuum near the earth\'s surface, is expressed very nearly by the \nequation s =16.1^\'^; s being in feet and i in seconds. Construct a table \ngiving the entire distance fallen through in i second; in 2 seconds; \nin 3 seconds; and in 4 seconds; the distance passed over during each \nof the above seconds; the velocity and acceleration at the end of each. \n\n\n\nTime in \nSeconds. \n\n\nEntire Distance \nin Feet. \n\n\nDistance each \nSecond. \n\n\nVelocity. \n\n\nAcceleration. \n\n\nI \n2 \n\n3 \n4 \n\n\n16.I \n\n64.4 \n\n144.9 \n\n257-6 \n\n\n16.I \n\n48.3 \n\n80.5 \n\n112. 7 \n\n\n32.2 \n\n64.4 \n\n96.6 \n\n128.8 \n\n\n32.2 \n32.2 \n32 2 \n32.2 \n\n\n\n128 DIFFERENTIAL CALCULUS, \n\n41. Having j- = 5/^, find the velocity and acceleration when / = \nseconds; if = 3 seconds. Ans, F\'i; = 2 = 3\'/io/2. Vt=2>= sV^S/\'^- \n\nAt = 2 = 31/572/4. ^^=3 = 31/573/4. \n\n\n\n42. _)/=sin (w sin-i jr). dy/dx\xe2\x80\x94m cos (w sin-^ \'^)/Vi \xe2\x80\x94 ^\'^\' Hence, \n\n(i - x\'\'){dyjdxf = m^ cos2 (m sin-i x) = m\\i -/). \nDifferentiating again and dividing by 2dy, we have \n(l - x^){d-^y/dx^) - x{dy/dx) + my = O. \n\n43. fx = sinh X. f"x = sinh ;f. \n\n44. fx = cosh X. /"x = cosh ;r. \n\n104. Leibnitz\'s Theorem \xe2\x80\x94 Let ji^ = uv, in which ?^ and v \n\nare any functions of x ; then (\xc2\xa7 75) \n\ndy/dx \xe2\x80\x94 2^ dv/dx + Z\' du/dx^ \nd\'^y/dx\'\' = u d\\^/dx\' + 2{du/dx)(dv/dx) + z\' d\'\'u/dx\\ \n\nd\'\'y _ d^ du d\'^v d\'^u dv d^u \ndx\' ~ ^dx\' ^^Jxd?~^^~d?dx~^ ^dx\'\' \n\nin which the numerical coefficients follow the law of those \nof the binomial formula. By a method similar to that used \nin deducing that formula for positive entire exponents it \nmay be shown that \n\ndx"" ~^ dx""^ ^dxdx\'\'-^\'^ 1.2 dx^ dx\'\' "\xc2\xbb + \' \' * \n\n,n{n\xe2\x80\x94\\)...{n \xe2\x80\x94 r^\\)d\'\'ud\'\'~\'\'v d\'^~\'^u dv d^u \n\n^ |7 di" dx^-^"- "^ \xe2\x80\xa2 * \xe2\x80\xa2 ^^.r"-!\' ^ "^ \'\'\'^\' \n\nEXAMPLES. \n\nI. f ^ /^"^X \n\nu \xe2\x80\x94 n. \n\n1, if ;;?=\xc2\xab \n00 , iim<.n. \n\n\n\n3. (sin x/x)^^^ = cos j;]^ = I.* \n\n4. (^--i)A]o = ^^]o=i. \n\n5. tan ^/^Jq = sec\'* xH = i. \n\n6. {x^ - a^)/ix\' - a^)\\ = 3^V2^]\xc2\xab = 3\xc2\xab/2. \n\n7. (^2 ~ ^2)/(^ _ ^f^^ - 2x/2{a - X)\\ = CO . \n\n8. {X - af\'/ia - x)y\\ = |(x - .)Vy|(;. - .)V4]^ = o. \n\n9. (i \xe2\x80\x94 sin x)/cos ^J^/2 = cos x/sin \xe2\x80\xa2^J^/g = o\xc2\xab \n\n10. (^^ - ^-^)/sin jc]^ = (^^ + ^-^)/cos jf]jj = 2. \n\n11. xVsin x]q = 2^/cos jt]^ = o. \n\nx^ I IX \'~\\ _ X cos ^ I _ cos ^ \xe2\x80\x94 ^ sin X I _i \nsin jr/ cos ;cJo 2 sin x Jo 2 cos x Jo~\'2\' \n\nX \xe2\x80\x94 sin x\\. _ I \xe2\x80\x94 cos jf I _ sin x~^ __ cos ^"1 i \n\nI \xe2\x80\x94 cos x\\ sin X I _ I \n\nsin\'* ^ _Jq 2 sin ;r cos jr_j0 2* \n\n* Hereafter, for abbreviation, the forms f{x)^^^ and fx~\\^ will fre- \n\nlimit , ^ ^ \nquently be used to express {fx). \n\n\n\nINDETERMINATE FORMS, \n\n\n\ni6i \n\n\n\n15. \n\n\n\n<\xc2\xbb*\xe2\x80\x94 2 sin X- \n\n\n\n;(? Jo I \xe2\x80\x94 cos X Jo \n\n-1 = \n\n\n\n^^-|-2 sin x\xe2\x80\x94e-=^ 1 \nsin jr Jo \n\n\n\ne^ -|- 2 cos jr + \n\n\n\n^mx _ ^ma n ^ ^^ mx "j _ ( 00 when r > I. \n\n^ \' {x - aj \\a ~ r(x \xe2\x80\x94 af^Ja " 1 o when r < 1. \n\nLimits of factors of the given or any derived ratio may be deter- \nmined separately (\xc2\xa7 36). \n\n\n\n\\/x tan X ~\\ k/^ "1 ^^" "^"l -^ \n\nThe given or any derived ratio may be separated into parts (\xc2\xa7 35). \n18. j/^- i/a-}-Vx-a ~] ^ / i_ ^ I \\ / -y "1 \n\n\n\n2x\\/x 2x\\/x \xe2\x80\x94 a. \n\n~ 2xr x-a Ja ~ ^Ta \ntan\xc2\xa3_\xe2\x80\x94 _sin^~| _ / sin x \\ /sec x \xe2\x80\x94 i\\~l _ sec ^ \xe2\x80\x94 il \n\nJo w A""^^^ yjo~ ^^ J( \n\n\n\nsec X tan x \n\n\n\nx^ I _ sec^x-|-tan\'*;r secjcj __i_ \nJo 2 Jo~2\' \n\ntan jf \xe2\x80\x94 sin or\\ /tan ^\\ /i \xe2\x80\x94 cos x\\~l i \xe2\x80\x94 cos or\\ \n"\xe2\x80\xa2 \xe2\x80\x94l?^ Jo = (\xe2\x80\x94 ) i--^-) J. = -7^ Jo \n__ sin x "1 cos X \n\n~(n- i)x\'\'-\'Jo ~(n- i)(n -2);r"-\'Jo "" "^ \' \n21. log (I +x)/x\']^ = 1/(1 + x)\\ = r. \n\nIn some cases it is advisable to transform the terms before apply- \ning the above rule. Thus \xe2\x80\x94 \n\nsin X "I 2 sin (x/2) cos (r/2)~l . . -, \n\n22. = ~- \xe2\x80\x94 -\xe2\x80\x94-- \xe2\x80\x94 = cot (x/2) L = 00. \n\nI \xe2\x80\x94 cos ^J 3 sin^ {x/2) ^ \' \'_[0 \n\n\n\n1 62 DIFFERENTIAL CALCULUS. \n\nii8. Form oo/oo , \n\nIf yiz = 00 = 0^, we may write \n\n\n\n~;r \xe2\x80\x94 ^r~r \xe2\x80\x94 ~-> and (\xc2\xa7117) \n\n0\xc2\xab *al fa o ^^ " \n\n\n\n0\xc2\xab 0^/ /< \nlimit r I / I "1 ,. r0\'^ //\'jt: "I \n\n^~.Ls/aJ="\'\xc2\xb0LwV7?J \n\n=-[(i)\'(g)]- \n\nand since limits of equimultiples of two variables with \nequal limits are equal, we have, multiplying by \n\n o as x ^-\xc2\xbb 00 , we have \n\nIn some cases we may with advantage place x \xe2\x80\xa2= a \xe2\x96\xa0\\- h and sub- \nsequently make ^ = o. Thus ^x \xe2\x80\x94 a/ \\/x^ \xe2\x80\x94 a^ reduces to 0/0, and \nthe ratios of all derivatives of both terms become 00/00 when x = a; \nbut putting jc = a -|- ^, we obtain \n\n\\/~^^^^a "I _ /^V3 n _ /?i/i^ ~] _ \n\n\n\nGO. \n\n\n\n164 DIFFERENTIAL CALCULUS, \n\n%. \n\n119. Form 0.00 . \n\nIf /^ = o and 0^ = 00 , we write \n\nwhich takes the form 0/0 or 00 /co when x = a^ and the \nlimit may be determined by the method of \xc2\xa7\xc2\xa7117, 118 \nwith the same limitations. \n\n\n\nEXAMPLES. \n\n-\'" ;r Jo \xe2\x80\x94 X Jo \n\n2. //^\'^ = \xe2\x80\x94 = = CO . \n\nJo x-^X 2 Jo \n\n3. .-VV],=;.V^V-],=0. \n\n4. e-^ log x]^ = log x/e\'\'\\ = (i A)A^^ = O. \n\n5. sin jir log cot x\\ = (sin jr/x) ;v log cot x\\ = log cot jf/(i/;i;)T \n\n= (x\'^/sin^^)(i/cot x)^^ \xe2\x80\x94 o. \n\n= I \xe2\x80\x94 \\j/{x^ + i)]^ = I, when tan-i(i/oo) = O. \n\n7. sec J? (:v sin x\xe2\x80\x94 7t/2)\'J^^^ = (;r sin x \xe2\x80\x94 7t /2)/cos \xe2\x96\xa0^J^/2 \n\n= (a- cos jc -{- sin .jf)/ \xe2\x80\x94 sin x \\, = \xe2\x80\x94 i. \n\n8, log (2 - i), tan ^]^ = log(a - ^) /cot ^]^ \n\n\xe2\x80\x94 \xe2\x80\x94 I / \xe2\x80\x94 7t/2a ~| _ \xc2\xa3 \n\n~~ a(2 \xe2\x80\x94 x/a)/ sin^(7rx/2\xc2\xab) J^ ~ it\' \n\n\n\nINDETERMINATE FORMS. 1 65 \n\n\n\n10. x\'^{\\ogxf\\^ \n\n\n\n\xe2\x80\x94 \xc2\xab/x(log xf\'^^_\\ \n\n\n\n.], \n\n\n\ntnx \ni/(log xfA( \n\nwhich remains indeterminate under the method; but placing x \xe2\x80\x94 e~^, \nwhence x^{\\og xf = (\xe2\x80\x94 ify\'/e"-^ and ^/m-^co as x-m-^o, we have \n(example 7, \xc2\xa7 iiS) o for the limit. \n\n120. Form 00 \xe2\x80\x94 00 . \n\n\\i fa \xe2\x80\x94 *

K.n^)) dz J \n\n+ ..., \n\nwhich is called Laplace\'s Theorem. \n\nSince the theorems of Lagrange and Laplace depend upon \nthat of Stirling\'s, they hold only when x is small enough to \nmake the developments convergent. \n\nEXAMPLES. \n\nI. Develop ;j/ = z -\\- xe** \n\nIn this case /( r) = y, f{z) \xe2\x80\x94 z, \n\n(p(y) = e^ , and (p{z) = /. \n\nHence, from Lagrange\'s theorem, we obtain \n\n\n\n2 \n\n\n\n13. \' \' |\xc2\xab \n\n\n\n202 DIFFERENTIAL CALCULUS. \n\n2. Given log^ = ;rv, develop^. \n\nWe may write y = e=>^y, and putting xy^=:y\\ we have y = xe^ \\ \nwhich may be developed by making 2 = and y =y in example i\xc2\xab \ngiving \n\nReplacing y by xy and dividing by x, we have \n\n3. Develop y = z -{- xy^. \n\nHere (<\')>P\'(a). \n\nHence, to obtain f"{a), multiply the differential coefficient \nof that factor of f\\x) which reduces to o by the other factors, \nand substitute a. \n\nTo illustrate, let \n\nf{x) = (x- \xc2\xab) V. \n\nf\'(pc) = 2x(x \xe2\x80\x94 a)(2x \xe2\x80\x94 a) =^ o gives \n\nX ^ o, X = a, X = a/2. \n\ny"(o) = 2(x \xe2\x80\x94 a)(2x \xe2\x80\x94 a)o = 2^^ indicating a minimum. \n\nf"(a) = 2jc(2ji; \xe2\x80\x94 cL)a = 2a^, indicating a minimum. \n\nf"(a/2) = /\\x{x \xe2\x80\x94 a)^,^ = \xe2\x80\x94 d^, indicating a maximum. \n\nEXAMPLES. \n\nJ^ind the values of the variable which correspond to maxima \nor minima of the following functions : \n\n(x -f- 3)^ X = o, min. \n\n\' (jc + 2)* \' X ^ \xe2\x80\x94 2, max. \n\n{y \xe2\x80\x94 \\f J = I, min. \n\nQ/ -f- 1)=^* Jf\' = 5, max. \n\n\n\n220 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n3, 1/^2^2 _ ^4. \n\n\n\n4. \n\n\n\n2r;rj[:\' \xe2\x80\x94 TtX^ \n\n\n\n2.r \xe2\x80\x94 ^3. \n\n\n\n5. a*x \xe2\x80\x94 X \n\n6. aj^\' \xe2\x80\x94 ;\xc2\xab:*, \n\n\n\n^ = o, min. of power. \n\xe2\x80\xa2\xc2\xab\xe2\x96\xa0 = \xc2\xb1 <^/V\'2, max., min. \n^ = o, min. \n^ = 4r/3, max. \n\nX = a/4/3, max. \nX = 3rt;/4, max. \n\n\n\n7. x-" - X \n\n\n\n5/2. \n\n\n\n8. \n\n\n\nX^ - 2^" \n\n\n\n2^" + 8 \n9. x\'^. \n\n10 (2^x< + a^bx)/a^. \nb \n\n\n\n^ 1 \n\n\n\n{c - xf \n\n\n\nb a \n\n\n\nX = o, mm. \n\nX = 16/25, max. \n\nX = o, max. \njf = 1. 19, min. \nX = i/e, min. \n\nX = \xe2\x80\x94 a/2, min. \n\n^ ya \n\nX \xe2\x80\x94 \xe2\x80\x94 Y= 3"^^\xc2\xbb min. \n\nj/^ _|_ |/^ \n\n6 = tan-i |/3/dz, min. \n\n\n\nsin cos \n\n13. fx = x/2 \xe2\x80\x94 x^ sin {i/x)f2. \n\nfx \xe2\x80\x94 1/2 + cos (i/x)/2 \xe2\x80\x94 X sin (i/x). \n\n1,1 i-i I .1 I \n1 cos \xe2\x80\x94 = cos\' \xe2\x80\x94 X sin \xe2\x80\x94 = 2jr sm \xe2\x80\x94 . cos \xe2\x80\x94 . \n\n2 2 ;f 2X X 2X 2X \n\nHence, fx = cos \xe2\x80\x94 . ( cos 2x sin 1 = o gives \n\n2X \\ 2;\xc2\xbbr 2x1 \n\n\n\nM \n\n\n\n\n\n2X \n\n\n\\ 2X \n\n\n2XJ \n\n\n\n\nX \n\n\n= 1/^, \n\n\nJT = 00. \n\n\nf\\i/n) = \n\n\n- 4/^^ \n\n\nindicating a maximum. \n\n\na-^x \n\n\n\n\n\n\nX \xe2\x96\xa0= \xe2\x80\x94 a, min. \n\n\n{a - xf \xe2\x96\xa0 \n\n\nX = ay max.. \n\n\nab \n\n\n._./\xc2\xab= + \'\'\xe2\x96\xa0\' \n\n\n\nX Va" -^ b"" - x"" \n\n\n\nMAXIMUM AND MINIMUM STATES. 22 \n\n\n\nPROBLEMS. \n\n1. Divide a number a into two such parts that the product of the \nmS^ power of one and the w*^ power of the other shall be a maximum. \n\nfx = x^^{a \xe2\x80\x94 xY, f\'x \xe2\x80\x94 x"^-\\a \xe2\x80\x94 xy^-\'^[?na \xe2\x80\x94 (w + n)x] = o \n\ngives X ^ o, X ^ a, x = ma /{in -\\- n), \n\n/"ma/{m -]-\xc2\xab)=: \xe2\x80\x94 {m -{- n)c, indicating a maximum. \n\n2. Divide a number a into two such factors that the sum of their \nsquares shall be a minimum. \n\nfx=:x\'^-\\- a\'^/x^, X =^ \xc2\xb1 ^a, minimum. \n\n3. Into how many equal parts must a number a be divided that \ntheir continued product may be a maximum? \n\nLet X = the number of equal parts, then \n\nfx \xe2\x80\x94 {a/xY, . *. log fx = x (log a \xe2\x80\x94 log x). \n\nf\'x = fx{\xe2\x80\x94 I -j- log a \xe2\x80\x94 log x) = o gives x = a/e, \nf"x{a/e) = W^(\xe2\x80\x94 e/a), indicating a maximum. \n\n4. Let A be the hypothenuse of a right triangle; find the lengths of \nthe other sides when the area is a maximum. \n\n\n\nLet X = one side, then \\/h^ \xe2\x80\x94 x^ = the other. \n\n\n\n/x = area = x r/i\'^ \xe2\x80\x94 x\'^/2. f\'x = o gives /^^ \xe2\x80\x94 ix\'^ = o, \nwhence x = hj V2, f\'\\h/ V2) \xe2\x80\x94 \xe2\x80\x94 ^/iK \n\n5. What fraction exceeds its \xc2\xab*^ power by the greatest number \npossible ? \n\nLet X = fraction, then fx=x \xe2\x80\x94 x^. \n\nf\'x = I \xe2\x80\x94 \xc2\xabjc\xc2\xab-i = o gives X = 1/ \\n. \n\n\xe2\x80\x9e( r-h-\\ I \xe2\x80\x94 \n\nf \\\\/ \\n)^^\xe2\x80\x94n{n\xe2\x80\x94\\)l \xc2\xab\xc2\xab~^ , indicating a maximum. \n\n6. Of all isoperimetrical rectangles which has the greatest area ? \n\n\n\n222 DIFFERENTIAL CALCULUS. \n\n7. On the right line A c B joining \n\nthe two lights A and B, find the point between the lights of least \nillumination. \n\' Let c = number of miles from A 10 B. \n\nLet X = number of miles from A to required point. \n\nLet a = intensity of the light A at i mile from A, \n\nLet d = intensity of the light ^ at i mile from B. \n\nThen \n\na b \n\nfx \xe2\x80\x94 -^-\\-7\xe2\x80\x94^ \xe2\x80\x94 C7is a minimum when = tan-i ^bja\'y \n\nOB X OCis a minimum when = tan-i(V\xc2\xab); \n\n0B-\\- OC-^ CB is a minimum when = tan-i ^ + ^^ ; \n\na -f- ^lab \n\nOB X OC X C!i5 is a minimum when \n\nla tan^ \xe2\x80\x94 5 tan\'^ + a tan \xe2\x80\x94 23 = o. \n\n16. Determine the maximum right cone which can be inscribed in \na sphere whose radius is A\'. \n\n\n\n224 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\n\nLet X = AF, and y = PB. \n\nThen \n\ntcv^x \nvol. = z/ = ~ \xe2\x80\x94 , but y^ = 2Fx \xe2\x80\x94 x\\ \n\nTherefore v = {2F7tx^ \xe2\x80\x94 7tx^)/3. \ndv/dx = nx{^R \xe2\x80\x94 3-^)/3 = o \ngives X = o, X = 4R/3. \n\nd^v/dx\'^ = 7r(4/? - 6^)/3 = - 4^V3]^=4;?/3. \n\n17. Find the radius of a circle such that the segment corresponding \nto an arc of a given length shall be a maximum. \n\nLet 20! = length of arc, and r = radius. a. \n\nDraw CD bisecting the arc, then \n\nZDCA = 6/2 = a/r, and = 2a/r. \n\nSegment = sector BCAD \xe2\x80\x94 aBCA \n\n= r\'9/2 - r"" sin 6/2 \n\nz=i ra \xe2\x80\x94 r^ sin {2a/r)/2, \n\nwhich is a maximum when r = 2a/7t, and \nthe segment is a semi-circle. \n\n18. With a given perimeter find the radius which makes the cor- \nresponding circular sector a maximum. Ans. radius = 1/4 perimeter. \n\n19. Find the maximum right cylinder which can be inscribed in a \ngiven right cone. \n\nLet VA = a, BA = b, AC = x, \nCD =y, CV = a \xe2\x80\x94 x. \n\nHence, vol. cylinder = z/ = Tty^x, \nVA -.ABwVC: DC, .\'. y= b{a - x)/a. \nTherefore v = nl)\'^{a \xe2\x80\x94 xfx/a^. \nOmitting nb\'^/a^, we have \n\nJ{x) = a^x - 2ax\'\' + x^. \n\n\n\n\n\nMAXIMUM AND MINIAiUM STATES. \n\n\n\n22 C \n\n\n\nf\\x) = a^ \xe2\x80\x94 ^ax -j- 3Jt^ = o gives x \xe2\x96\xa0= a or (7/3. \n/"[a/\'i) = \xe2\x80\x94 la, therefore v = /\\TTaP/2-i is a maximum. \n\n20. Circumscribe the minimum isos- \nceles triangle about the parabola j^=4\xc2\xabx. \n\nLet X = C/\' = ^ (9, 7 = PM, h = OD. \nThen \n\n^ _ A. \n\nBD\xe2\x80\x94{h + x)y/2x = {h-\\- x)\\/ax/x, \n\nand area A \n\n= {k -{-xf \'i/ax/x, X = h/2, min. \n\n21. Determine the minimum right \ncone circumscribing a given sphere. \n\nLet X = alt. = AD, y = radius of base, \nR = radius of sphere. \n\nThen V = vol. of cone = ny^x/^. \n\n\n\n\n\ny\\R:\\ s/x\'- -f / \\x - R. \nFrom vi^hich y"^ = R\'^x/{x \xe2\x80\x94 2R), \n\nand V = 7rR^x^/3{x - 2R). \nX = 4R, min. \n\n\\B \n\n\n\n22. Find the maximum parabola that \ncan be cut from a given right cone. \nLet AC = a, AB \xe2\x80\x94b, DC = x. \n\n\n\nThen AD = a \xe2\x80\x94x, DE= Via \xe2\x80\x94 x)x. \nAlso, a :x: : d : DG. .\' . DG\xe2\x80\x94 bx/a. \n\n\n\nParabola = \\b \\/ax\'\'^ \xe2\x80\x94 x\'^/^a, \nX = 3\xc2\xab/4, max. \n\n\n\n23. Find the maximum isosceles triangle inscribed in a given \ncircle. \n\n\n\n\n226 \n\n\n\nD IFFEREN TIA L CA L CUL US. \n\n\n\n\n]Let r \xe2\x80\x94 radius CA, AB \xe2\x80\x94 AE \xe2\x80\x94 x, BE = 2y. \nThen area A\xe2\x80\x94 u =^y ^ x^ \xe2\x80\x94 y^. \nAlso, u \xe2\x80\x94 \n\n\n\nAB y^ AE y^ BE \n\n\n\n4r \n\n\n\nx\'^y \n\n\n\nx\\/4) \n\n\n\nAnd u \n\n\n\ni/Ar^- X\' \n\n\n\n. X = ^4/3, max. \n\n\n\n2r ir \n\n24. Find the maximum cylmder that can be inscribed in a given \nprolate spheroid. \n\nLet Q.X \xe2\x80\x94 axis, and y \xe2\x80\x94 radius of \nbase, of required cylinder. \nThen, vol. of cylinder \n\n= z/ = 2iiy\'^x \xe2\x80\x94 27txb\'^{a\'* \xe2\x80\x94 x\'^)/a^, \nwhich is a maximum for x \xe2\x80\x94 (2/ 4/3. \n\n25. Find the minimum isosceles triangle circumscribing a given \ncircle. \n\nLet y = radius CE, x \xe2\x80\x94 BF^ \n\n2y - AD. \n\nThen area A = xy. \n\nSimilar triangles, BCE, BED, give \n\ny : y : ^ x\'\' -\\- y^ : x - r. \n\n\n\n\n\nHence, y - r\\/x/{x - 2r), and \n\n\n\narea ^ - xr \\/x/{x - 2r), \nD which is a minimum for x = 3^. \n\n26. " A boatman 3 mi from shore goes to a point 5 \nmi. down the shore in the shortest lime. He rows 4 B_ \nmi. and walks 5 mi an hour, Where did he land? \n\nLet B be the boat 3 mi. from S, vvhich is 5 mi. \nhorn the point /*. Let IV be the landing-place, and \n\nThen, number of hours \n\nr- i ^ 1/9+^74 -i- (5 - x)/s, \nwhich is a minimum for jc = 4. \n\n\n\n*Todhunters Difl. Calc, p. 213. \n\n\n\nMAXIMUM AND MINIMUM STATES. \n\n\n\n227 \n\n\n\n27. Through a given point P within an angle \nBAC draw a right line so that the triangle \nlormed shall be a minimum. D \n\nDraw PD parallel to AC, and let AD = a. \n\nPD = h, AX - X. ,\'. DX =: X \xe2\x80\x94 a. \nThen x \xe2\x80\x94 a\\b \\\\ x : AR. \n\nAXR \xe2\x80\x94 X AR-sAw A J 2., which is a minimum for x \xe2\x80\x94 2a. \n\n\n\n\nXP=PR. \n\n\n\n\n28. The volume of a cylinder being constant, find its form vyhen the \nsurface is a mmimum. Ans. Altitude = diameter of base. \n\n29. Find the height of a light A above the \nstraight line OB when its intensity at ^ is a \nmaximum. \n\nLet a = intensity of light at i foot from the \nlight. \n\nB OA - y, OB-b, I OB A - 6. \n\nThe intensity varies directly as sin 9, and mversely as BA . \nIntensity at B = ay/{b^ -|- y^f^ , which is a maximum when \n\ny = d \\/2 1 2. \n\n30, Having y =^ x tan a \xe2\x80\x94 x\'^/^h cos\'^ a . 1\xc2\xb0. Find the maximum \nvalue of y. 2*. Considering/ = o and a as varying, find maximum \nvalue of X. \n\n1st. y \xe2\x80\x94 h sin- a, a maximum, x = h sin 2a. \n\n\n\nAns. \n\n\n\n2d. X \xe2\x80\x94 \xe2\x96\xa0 2h, a maximum, a = 45\' \n\n\n\n31. On the right line CC\\ -- a joining the centres of two spheres \n(radii A, r) find point (rom which the maximum spherical surface Is \nvisible. \n\n\n\n\n22S DIFFERENTIAL CALCULUS. \n\nLet CP = X. .\'. PCx - a - X. \n\nArea zone ASHB = inR X HS. \n\nX- Rv. R: CS. .-. CS := R^x and HS-R- R^/x. \nHence, zone ASBH = 2;rA\'(A\' - R\'\'/x). \nSimilarly, zone DEL \xe2\x80\x94 2ni(r \xe2\x80\x94 r\'^/{a \xe2\x80\x94 x)\\. \n\nVisible surface --^ 27t\\^R\'\' + r^ - (^R^x + rV(\xc2\xab - ^))], which is \na maximum for x = aR^^\'^/^R^^^^ + r^^% \n\n32. Find the path of a ray of light from \na point /i in one medium to a point B in \nanother medium, such that a minimum \ntime will be required for light to pass \nfrom A to B; the velocity of light in the \nfirst medium being V, and in the second \nv. [Fermat\'s Problem.] \n\nIt is assumed that the required path is \nin a plane through A and B perpendicu- \nlar to the plane separating the media. \n\nLet ACB be the required path. Through A, C, and B draw per- \npendiculars to DE. \n\nLet a - AE, b = DB, d- DE, \n\nThen AC = \xc2\xab/cos 0, BC = b/cos (p\' , \n\nCE = a tan 0, CD \xe2\x80\x94 b tan (p\' . \n\na tan (p -\\- b tan 0\' \xe2\x96\xa0= d. .\'. d \'\' \nand a=^ + <5\' = f^ Then [Anal. Geom.]y^+y2 = t^ \n\n\n\n. \xe2\x80\xa2. / = V^2 _ ^\'2 and J = y + V^^ \xe2\x80\x94 x"^. \n.*. ds/dx\' = I \xe2\x80\x94 x\' I Vc^ \xe2\x80\x94 x"^ = o gives x\'^ \xe2\x80\x94c\'^/i =y2. \n\nThat is, equal conjugate diameters are those whose sum is a maxi- \nmum. \n\nExpressing x\' and y in terms of the inclination of x to the trans- \nverse axis, denoted by 6, we have ds/dB = {ds/dx\')(dx\'/dQ). \n\na and b are, respectively, maximum and minimum states of x\\ \ngiving dx\' /dB \xe2\x80\x94 o, and therefore ds/dB = o. Hence the sum of the \naxes is a minimum. \n\n37. " A rectangular hall 80 feet long, 40 feet wide, and 12 feet high \nhas a spider in one corner of the ceiling. How long will it take the \nspider to crawl to the opposite corner o n the floo^ if he crawls a foot \nin one second on the wall and two feet in a second on the floor?"* \n\nAns. 55.4754 seconds, minimum. \n\n1 39* ^^ find the maximum and minimum distances from \na given plafie curve to a given point in its plane. \n\nLet y = fx be the equation of any plane curve, (a, b) \nthe coordinates of any point in its plane, and i? the dis- \ntance from [a, b) to the point {x\\y\') on the curve. \n\n* Problem proposed by Professor H. C. Whitaker in American \n\nMathematical Monthly, Vol. L No. 8. \n\n\n\nMAXIMUM AND MINIMUM STATES. 23 1 \n\n-If {x\' ^y\') move along the curve, R in general becomes a \nvarying distance measured on the radius vector joining (^, b) \nwith the moving point {x\' ^y\') and i?" =(a:\' \xe2\x80\x94 of + (jv\' \xe2\x80\x94 ^)^ \nWe wish ,to find the maximum and minimum values of R. \n\nPlacing the first derivative of {x\' \xe2\x80\x94 ay -\\- (y\' \xe2\x80\x94 by equal \nto zero, we have \n\n^\' _ ^ _l- (/ _ b){dy/dx\') = o. . . . (1) \n\nThe equation of the normal to y ^= fx at {x\',y\') is \ny \xe2\x80\x94 y\' = \xe2\x80\x94 {dx\'/dy\')(x \xe2\x80\x94 x\'),^i^S. Hence (i) expresses \nthe condition that (x\',y\') is on the normal through {a, b). \n\nThe required value of R is therefore estimated along the \nnormal through {a, b), and is a maximum or a minimum \naccording as the second derivative (dropping the primes), \n\nI + (Jy/dxY + (/ - i){dy/dx\'), \n\nis negative or positive, and, in general, is neither a maxi- \nmum nor a minimum when the second deiivative reduces \nto zero (\xc2\xa7 136). \n\nAs {a, b) may be any point upon any normal, we con- \nclude that the radial distance of each point of a normal \nfrom the curve is, in general, a maximum or a minimum \nwhen measured upon the normal. (See figure, page 232.) \n\nThus, let BAM be a normal to the curve NMO at M. \n^With A and B as centres, and with the radii AM a.nd BM \nrespectively, describe the circumferences rMr and RMR. \nThe figure shows that the radial distance of A from NMO \nis a minimum when measured upon the normal AM, and \nthat the corresponding distance of the point ^ is a maxi- \nmum. This is evident from the fact that the circumference \nvMr^ in the vicinity of and on both sides of M, lies within \nthe curve NMO, while the corresponding part of the cir- \ncumference RMR lies without. \n\n\n\n232 \n\n\n\nDIFFERENTIAL CALCULUS. \n\n\n\nConsider the point [a^ b) to move upon the normal, and \nlet yoc^yj be its variable coordinates. When the normal \n\n\n\n\ndistance of \\Xyy ) from the curve is neither a maximum nor \na minimum, we have \n\nI + {dy/dxY + (y --y){d-^y/dx\') = o, \n\nwhence by combination with (i) we obtain \n\n\n\ndx\' \n\n\n\n\\ dx^ jdx \' dx \n\n\n\n(2) \n\n\n\nfor the coordinates of a point on the normal whose distance \nfrom the curve measured along the normal is, in general, \nneither a maximum nor a minimum. Representing this dis- \ntance by p, we have \n\np^ = {x-Tf + {y-y)\\ \n\nwhich combined with (2) gives \n\n\n\np= 1 + \n\n\n\ndx\'\' \n\n\n\n\'\xe2\x96\xa0 l<\xc2\xa3l \n\nI dx\' \n\n\n\n(3) \n\n\n\nMAXIMUM AND MINIMUM STATES. 233 \n\nIn the figure, page 232, \\x^y ) lies somewhere between A \nand B. It separates those points of the normal each of \nwhich has a minimum radial distance from the curve lying \non the normal, from those points of the normal each of \nwhich has a corresponding maximum distance on the same \nline. \n\nIt is important to observe that a circumference described \nwith \\x, J\' ) as a centre and with a radius equal to p will, \nin general, intersect the curve NMO at M. \n\nThis circle is important in the discussion of curves, and \nequations (2) and (3) will be referred to hereafter. \n\nIMPLICIT FUNCTIONS. \n\n140. Having J given as an implicit function of x^ by an \nequation f{^x^y) = o not readily solved with respect toj, \nwe may differentiate as indicated in \xc2\xa7 no and obtain an \nexpression for dy/dx. Placing it equal to o, we may com- \nbine the resulting equation with the given, and find critical \nvalues of x. \n\nOtherwise, let u=f{x^y) = o (i) \n\nThen (i), (\xc2\xa7111), \n\ndu/dx = du/dx -f (di^/dy) (dy/dx) = o. . (2) \n\nMaxima and minima values of y in general require \ndy/dx = o. Hence, \n\n^ \xe2\x80\x94f(^,y) \xe2\x80\x94 o, combined with \'du/dx = o, \\ \n\ngives critical values of x. \nEq. (6) {\xc2\xa7 in) gives \n\nd\'^y/dx^ = - (dW^^l/idu/dy), \n\nwhich, if not zero or infinity, is positive for a minimum and \nnegative for a maximum oi y. \n\n\n\n234 DIFFERENTIAL CALCULUS. \n\nHaving y \xe2\x80\x94 fZj z = cpx, \n\ndy/dx = (dy/dz) X (dz/dx) = o, \nwill give critical values of x. \n\nEXAMPLES. \nI. u = x^ -\\- y^ \xe2\x80\x94 2^^^ = o\xc2\xab \n\ndu/dx = sx^ \xe2\x80\x94 Sa"^ = o. .\'. X = \xc2\xb1 a. \nSubstituting in given equation, we havejj/ = \xc2\xb1 a ^2. \n\'d^ujdx^ = tx, \'du/\'dy = s^y". \n\n^\'Vn -6a 3/-. \n\n:74 \\x=a = , 3/- > \xe2\x80\xa2 \xe2\x80\xa2 ji\' = \xc2\xab y2 IS a maximum. \n\n\n\n^vn 6a 8/\xe2\x80\x94. . . \n\n\xe2\x80\x94 ^n, . . jj/ = \xe2\x80\x94 a y 2 IS a mmimum. \n\n\n\n\'1 - - \xe2\x80\x94 \n\n2. Jf^ \xe2\x80\x94 saxy -j-jj/^ = o. jr = o, jj/ = o, is a minimum. \n\nX = a 4/2, y \xe2\x80\x94 a I/4, is a maximum \n\n\n\n3. jc" -j-^\'\' \xe2\x80\x94 23xy \xe2\x80\x94 a\' = o. \na3 \n\n\n\n^ = , IS a maximum. \n\n\n\n4. 4XJJ\' \xe2\x80\x94 ;j/4 \xe2\x80\x94 jc* = 2. X = \xc2\xb1 I, >\' = \xc2\xb1 I, no max. or min. \n\n5. jv\'^ \xe2\x80\x94 3 = \xe2\x80\x94 2Jr(jj/x + 2). X = \xe2\x80\x94 1/2, J = 2, a maximum. \n\n6. jj/ = Tt^zjuy z r= {k"^ -\\- x\'^)/x, X z= k makes ^ a minimum. \n\n141. Having v given as an implicit function of x^ by two \nequations v = (p[x^y) and ti ^= f{x,y) = o, from which _y \nis not readily eliminated, we may proceed as follows : \n\ndv _\'dv dv dy \ndx dx \'dy dx \' \n\n\n\ndy \'du fdu ( \\ ro X \n\n\n\ndx dxl \'dy \n\n\n\nMAXIMUM AND MINIMUM STATES. 235 \n\n\' dx ^ dx dy dx I Qy \' \n\ndv . dv 9^/ dv du , . \n\nand ^=0 gives -\xe2\x80\x94 _---=o, . . (i) \n\nwhich combined with u = /(x, y) = o gives critical values \nof X. The sign of the corresponding value of d\'^v/dx^ will, \nin general, determine whether z/ is a maximum or a minimum. \n\nEXAMPLES. \n\nI. 2/ = x\xc2\xab+/, (;\xc2\xbb: - \xc2\xab)\' -\\-{y- bf - c^ = 0. \n\ndv/dx \xe2\x80\x94 2x, dv/dy = 2y, \n\ndu/dx = 2{x \xe2\x80\x94 a), du/dy \xe2\x80\x94 2{y \xe2\x80\x94 5). \n\nSubstituting in (i), we have ay z=dx; which combined with m = o \ngives \n\n\n\nX = a \xc2\xb1ac/ Vd" -h b"". \n\nThe positive sign gives a maximum, and the negative a minimum, \nfor V. \n\n2. Find the points in the circumference of a given circle which are \nat a maximum or minimum distance from a given point. \n\n3 Given the four sides of a quadrilateral, to find when its area is a \nmaximum. \n\nLet a, b, c, d be the lengths of the sides, (p the angle between a and \nb, rj) that between c and d. \n\nThen area r=. v = ab sin 0/2 -|- cd sin ^/2, \n\nand a" _|_ ^2 _ 2^(5 cos (p = c^ ^ d"^ \xe2\x80\x94 2cd cos ^, \n\neach member being the square of the same diagonal. \n\ndv ab , dv cd \n\n-\xe2\x80\x94 = \xe2\x80\x94 cos 0, \xe2\x80\x94 \xe2\x80\x94 = \xe2\x80\x94 cos tp, \n\nd(p 2 ^\' a^ 2 \n\n-7- = 2\xc2\xab<5 sm 0, \xe2\x80\x94 = \xe2\x80\x94 2cd sm z^. \n\n^0 d^i^ \n\n\n\n236 \n\n\n\nDIFFERENTIA L CA L CUL US. \n\n\n\nSubstituting in (i), we have \n\ntan (p \xe2\x80\x94 \xe2\x80\x94 tan ^. . ". (j) \xe2\x80\x94 180\xc2\xb0 \xe2\x80\x94 ^. \nThat is, the quadrilateral is inscribable in a circle. \ndv ab ^ , cd , d^ \n\n\xe2\x80\x94 - = cos (p \\ cos Tp\xe2\x80\x94- \xe2\x80\x94 O, \n\nd(p 2 \'2 d(f) \n\nab sin (f)=^ cd sin \'^{dip/d(p) ; \nfrom which dTp/d(p = ab sin cp/cd sin ip. \n\nSubstituting in above, we have \n\ndv/d