The differential of any power of a function is equal to the exponent multiplied by the function raised to a power less one, multiplied by the differential of the function, d("") = nun - Idu. Formulas for differentiating algebraic functions. To find the differential of the form u = (a + bxn)m: dx. Multiply the exponent of the parenthesis into the exponent of the variable within the parenthesis, into the coefficient of the variable, into the binomial raised to a power less 1, into the variable within the parenthesis raised to a power less 1, into the differential of the variable. To find the rate of change for a given value of the variable: Find the differential coefficient, and substitute the value of the variable in the second member of the equation. du = 3x2. Example.—If x is the side of a cube and u its volume, u = x3, äx Hence the rate of change in the volume is three times the square of the edge. If the edge is denoted by 1, the rate of change is 3. Application. The coefficient of expansion by heat of the volume of a body is three times the linear coefficient of expansion. Thus if the side of a cube expands .001 inch, its volume expands .003 cubic inch. 1.0013 1.003003001. A partial differential coefficient is the differential coefficient of a function of two or more variables under the supposition that only one of them has changed its value. A partial differential is the differential of a function of two or more variables under the supposition that only one of them has changed its value. The total differential of a function of any number of variables is equal to the sum of the partial differentials. If u = f (xy), the partial differentials are du du dx, dy. dx + -dy + -dz; = 2xdx+3y2dy - dz. dy dz Integrals.-An integral is a functional expression derived from a differential. Integration is the operation of finding the primitive function from the differential function. It is indicated by the sign f, which is read "the integral of." Thus f 2xdx = x2; read, the integral of 2xdx equals x2. To integrate an expression of the form mxm-1dx or xdx, add 1 to the exponent of the variable, and divide by the new exponent and by the differential of the variable: 3x2dx = x3. (Applicable in all cases except when - 1 a = −1. Forfæ ̄ dx see formula 2 page 78.) The integral of the product of a constant by the differential of a variable is equal to the constant multiplied by the integral of the differential: The integral of the algebraic sum of any number of differentials is equal to the algebraic sum of their integrals: 2 b Z3 du = 2ax2dx – bydy — z2dz; fdu = ax3 — y2 3' Since the differential of a constant is 0, a constant connected with a variable by the sign + or - disappears in the differentiation; thus d(a+xm): = mx dx. Hence in integrating a differential expression we m dzm m-1 annex to the integral obtained a constant represented by C to compensate for the term which may have been lost in differentiation. Thus if we have dy: = adx; fdy = afdr. Integrating, y=ax ± C. The constant C, which is added to the first integral, must have such a value as to render the functional equation true for every possible value that may be attributed to the variable. Hence, after having found the first integral equation and added the constant C, if we then make the variable equal to zero, the value which the function assumes will be the true value of C. An indefinite integral is the first integral obtained before the value of the constant C is determined. A particular integral is the integral after the value of C has been found. A definite integral is the integral corresponding to a given value of the variable. Integration between limits.-Having found the indefinite integral and the particular integral, the next step is to find the definite integral, and then the definite integral between given limits of the variable. The integral of a function, taken between two limits, indicated by given values of x, is equal to the difference of the definite integrals corresponding to those limits. The expression is read: Integral of the differential of y, taken between the limits x' and x": the least limit, or the limit corresponding to the subtractive integral, being placed below. Integrate du = 9x2dx between the limits = 1 and x = 3, u being equal to 81 when x = 0. fdu = f9x2dx 3x3 + C; C = 81 when x = Ö, then x = 3 = du=3(3)3 +81, minus 3(1)3 +81 = 78. Integration of particular forms. To integrate a differential of the form du = (a + bx1)mxn - 1dx. 1. If there is a constant factor, place it without the sign of the integral, and omit the power of the variable without the parenthesis and the differ ential; 2. Augment the exponent of the parenthesis by 1, and then divide this quantity, with the exponent so increased, by the exponent of the parenthesis, into the exponent of the variable within the parenthesis, into the coefficient of the variable. Whence Sdu = (a+bxn)m+ (m + 1)nb 1 = C. The differential of an arc is the hypothenuse of a right-angle triangle of which the base is dx and the perpendicular dy. If z is an arc, dz = √dx2+dy2 z=S √dx2 + dy2, Quadrature of a plane figure, The differential of the area of a plane surface is equal to the ordinate into the differential of the abscissa. To apply the principle enunciated in the last equation, in finding the area of any particular plane surface: Find the value of y in terms of x, from the equation of the bounding line; substitute this value in the differential equation, and then integrate between the required limits of x. Area of the parabola. Find the area of any portion of the common parabola whose equation is Substituting this value of y in the differential equation ds = ydx gives 2 If we estimate the area from the principal vertex, x = 0. y = 0, and C = 0; and denoting the particular integral by s', s' = xy. That is, the area of any portion of the parabola, estimated from the vertex, is equal to 2% of the rectangle of the abscissa and ordinate of the extreme point. The curve is therefore quadrable. Quadrature of surfaces of revolution. -The differential of a surface of revolution is equal to the circumference of a circle perpendicular to the axis into the differential of the arc of the meridian curve. in which y is the radius of a circle of the bounding surface in a plane perpendicular to the axis of revolution, and x is the abscissa, or distance of the plane from the origin of coördinate axes. Therefore, to find the volume of any surface of revolution: Find the value of y and dy from the equation of the meridian curve in terms of x and dx, then substitute these values in the differential equation, and integrate between the proper limits of x. By application of this rule we may find: The curved surface of a cylinder equals the product of the circumference of the base into the altitude. The convex surface of a cone equals the product of the circumference of the base into half the slant height. The surface of a sphere is equal to the area of four great circles, or equal to the curved surface of the circumscribing cylinder, Cubature of volumes of revolution.-A volume of revolution is a volume generated by the revolution of a plane figure about a fixed line called the axis. If we denote the volume by V, dV = πу2 dx. The area of a circle described by any ordinate y is y2; hence the differential of a volume of revolution is equal to the area of a circle perpendicular to the axis into the differential of the axis. The differential of a volume generated by the revolution of a plane figure about the axis of Y is πx2dу. To find the value of V for any given volume of revolution : Find the value of y2 in terms of a from the equation of the meridian curve, substitute this value in the differential equation, and then integrate between the required limits of x. By application of this rule we may find: The volume of a cylinder is equal to the area of the base multiplied by the altitude. The volume of a cone is equal to the area of the base into one third the altitude. The volume of a prolate spheroid and of an oblate spheroid (formed by the revolution of an ellipse around its transverse and its conjugate axis respectively) are each equal to two thirds of the circumscribing cylinder. If the axes are equal, the spheroid becomes a sphere and its volume = 1 × D = ¦ The volume of a paraboloid is equal to half the cylinder having the same base and altitude. The volume of a pyramid equals the area of the base multiplied by one third the altitude. Second, third, etc., differentials. The differential coefficient being a function of the independent variable, it may be differentiated, and tre thus obtain the second differential coefficient: du d2u = Dividing by dx, we have for the second differential coeffi d2u cient which is read: second differential of u divided by the square of the differential of x (or dx squared). d3u The third differential coefficient is read: third differential of u divided dx3 by dx cubed. The differentials of the different orders are obtained by multiplying the d3u differential coefficients by the corresponding powers of dx; thus dx3= dx3 third differential of u. Sign of the first differential coefficient.-If we have a curve whose equation is y = fx, referred to rectangular coördinates, the curve dy will recede from the axis of X when is positive, and approach the dx axis when it is negative, when the curve lies within the first angle of the coördinate axes. For all angles and every relation of y and x the curve will recede from the axis of X when the ordinate and first differential coefficient have the same sign, and approach it when they have different signs. If the tangent of the curve becomes parallel to the axis of X at any point = 0. If the tangent becomes perpendicular to the axis of X at any dy dx Sign of the second differential coefficient. The second differential coefficient has the same sign as the ordinate when the curve is convex toward the axis of abscissa and a contrary sign when it is concave. Maclaurin's Theorem.-For developing into a series any function of a single variable as u = A + Bx + Сx2 + Dx3 + Ex4, etc., in which A, B, C, etc., are independent of x: In applying the formula, omit the expressions x = 0, although the coefficients are always found under this hypothesis. Taylor's Theorem. For developing into a series any function of the sum or difference of two independent variables, as u' = √(x + y): du du' in which u is what u' becomes when y = 0, is what becomes when dx y = 0, etc. Maxima and minima.-To find the maximum or minimum value of a function of a single variable: 1. Find the first differential coefficient of the function, place it equal to 0, and determine the roots of the equation. 2. Find the second differential coefficient, and substitute each real root, in succession, for the variable in the second member of the equation. Each root which gives a negative result will correspond to a maximum value of the function, and each which gives a positive result will correspond to a minimum value. EXAMPLE. To find the value of x which will render the function y a maximum or minimum in the equation of the circle, y2 + x2 = R2; mum for R positive. In applying the rule to practical examples we first find an expression for the function which is to be made a maximum or minimum. 2. If in such expression a constant quantity is found as a factor, it may be omitted in the operation; for the product will be a maximum or a minimum when the variable factor is a maximum or a minimum. 3. Any value of the independent variable which renders a function a maximum or a minimum will render any power or root of that function • maximum or minimum; hence we may square both members of an eo tion to free it of radicals before differentiating. By these rules we may find: The maximum rectangle which can be inscribed in a triangle is one whose altitude is half the altitude of the triangle. The altitude of the maximum cylinder which can be inscribed in a cone is one third the altitude of the cone. The surface of a cylindrical vessel of a given volume, open at the top, is a minimum when the altitude equals half the diameter. The altitude of a cylinder inscribed in a sphere when its convex surface is a maximum is r/2. r = radius. The altitude of a cylinder inscribed in a sphere when the volume is a maximum is 2r ÷ √3. (For maxima and minima without the calculus see Appendix, p. 1070.) Differential of an exponential function. The relation between a and k is ak =e; whence α = in which e = 2.7182818. . . the base of the Naperian system of logarithms. Logarithms.-The logarithms in the Naperian system are denoted by 1, Nap. log or hyperbolic log, hyp. log, or loge; and in the common system always by log. k = Nap. log a, log a = k log e. The common logarithm of e, = log 2.7182818 = .4342945 the modulus of the common system, and is denoted by M. Hence, if we have the Naperian logarithm of a number we can find the common logarithm of the same number by multiplying by the modulus. Reciprocally, Nap. com. log x 2.3025851. log If in equation (4) we make a = 10, we have 1 = k log e, or = log e = M. That is, the modulus of the common system is equal to 1, divided by the Naperian logarithm of the common base. From equation (2) we have If we make a = 10, the base of the common system, x = log u, and That is, the differential of a common logarithm of a quantity is equal to the differential of the quantity divided by the quantity, into the modulus. If we make a = e, the base of the Naperian system, x becomes the Na |