12. Find the length of the normal at the point (x = a) of the curve whose equation is x+4y = 4 a2. ASYMPTOTES TO CURVES. a Ans.√13. 52. If a curve have an infinite branch such that there is a straight line which it can never meet, but to which, if it be sufficiently extended, it can approach nearer than any distance, however small, that can be assigned, the straight line is said to be an asymptote to the curve. An asymptote is therefore to be considered as a line with which a tangent to the curve would continually tend to coincide, when one of the coordinates is increased without limit. Asymptotes parallel to either of the coordinate axes may be at once determined, by observing whether any finite value of one of the coordinates renders the other infinite. Thus if the equation of the curve be x=2r; therefore the curve has an asymptote parallel to the axis of y at If the value x = O renders y = ∞, the distance x = 2r from the origin. then the axis of y is an asymptote, and if y axis of x is an asymptote. = 0 gives x = ∞o, the Ex. Find whether the hyperbola has asymptotes; and, if so, determine their position. = The equation to the hyperbola is a2 y b2x2-a2b2, the origin being at the centre, and its sub 53. The asymptotes of a curve may be found in the following manner: Let A B be an asymptote of the branch of a curve CD, which is not parallel to the axis of y; then if y = ƒ x be the equation of the curve, the equation of the asymptote will be of the form y = ax + ß. . . . (1). Now the difference between a x + ß and the ordinate of the curve corresponding to the same abscissa, must necessarily be a certain function of x, which may be called V, and which approaches zero, when x ap proaches, according as the branch under consideration extends indefinitely on the positive or negative side of the axis of y. The equation of this branch of the curve will be, therefore, from which we have only to determine a and B, when approaches tend towards zero according as x increases, and will vanish when x = ∞, so that a is the limit of the second member of the equation, and consequently that of the first also; hence. Having found a, we get from (1), y · limit of the second member, since V = (3). -α x = BV; hence ẞ is the O when x = ∞ ; therefore B = limit of (y − a x) . . (4). The equation of the asymptote is thus completely determined, and thence its position is known. Let the equation of the curve be reduced, if possible, to the form y = ax + B+ V x ̄1 + d x2 + etc.; then when is taken indefinitely great, the terms involving negative indices will vanish, and the equation to the infinite branch of the curve will be simply y = ax + B (5), which is the asymptotic equation, and designates a rectilinear asymptote. EXAMPLES. 1. Find the asymptotes of the curve a2 y2 — b2x2 Here y b √(x2-a2) b = = = ∞, gives (3) Again, y -αx= ± √(x2 - a2) = -x, which, when x = ∞, gives (4) Otherwise. The equation of the curve can be reduced to the form therefore y = a b + a b a Gab x2 + x is the equation of the asymptotes, as before. 2. Let the equation of the curve be y2 Here α = limit of = and B = limit of (y — ax) = limit of {√(x2+x) − x}; but consequently B, and the equations of the asymptotes are y = x + 1, and y = −x + 1. x2 (x+a) 3. Let the equation of the curve be y2 = x-a If x = a, then y =; hence there is an asymptote parallel to the ± (x + a + a2 x1 + etc.) Hence the equations to the other two asymptotes are Ans. The axis of y continued in the negative direction is one of the asymptotes, and the other passes through the origin, and makes an angle of 45' with the axis of x. 5. Find the asymptote of the curve y3 = x3 +a x2. 6. Find the asymptotes to the curve y3 = 3x2 — x3. Ans. y = −x+1. 7. Find the equations of the asymptotes to the curve whose equation Ans. x = b, x = 2b, and y = x + 3b. is y (x2-3b x + 2 b2) = x3. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES. 54. In many inquiries in physical science, functions depend on two or more variables, which are independent of one another; thus the path described by a planet depends not only on its distance from the sun and its projectile motion, but it is also influenced by the attractions of the other planets. Let u = f(x, y) denote a function of two independent variables, x and y; and let x be increased by h, and y by k; then if u' = ƒ (x+h, y + k), we have the latter form being obtained by adding ƒ (x+h, y) and subtracting it, and introducing the multipliers and divisors h and k. Now if h be diminished towards zero, the first fraction in the second member will du dx, or dx; and if both h and k be dx diminished towards zero, the other fraction will approach to the limit or dy. Also u'u will be changed to du; hence df(x,y) du dy, dy dy The first member, du, is called the total differential of u; du du du are du dx dy dy dy are called the partial differentials of u; and and called the partial differential coefficients of u.* Since the position of a point in space is in general referred to three rectangular coordinate planes, the equations of surfaces, etc., are of the forms f(x, y, z) = 0, or z = f(x, y); and in the application of the differential calculus it is seldom necessary to develop a function of more than two variables. Hence, to find the differential of a function of two independent variables, differentiate the given function first with respect to one of the variables, and then with respect to the other; the sum of these partial differentials will give the total differential; thus, if ux" sin y, then du nx-1 dx sin y+x" cos y dy; 55. If a function of two variables be differentiated successively, first * The denominators dr and dy are not to be considered as mere divisors of du; but that is the differential coefficient obtained by regarding r alone as variable, du dx du and the one found by supposing y alone as variable. dy with respect to one of the variables, and the result with respect to the other, the final result will be the same in whatever order the processes succeed one another. Thus 3y differentiated with respect to x gives 3xy dx, and this differentiated with respect to y gives 6xy dx dy. But if we first differentiate with respect to y, we get 2x3 y dy, and this, with respect to x, gives, as before, 6xy dx dy. IMPLICIT FUNCTIONS. 56. When the function of x and y is an implicit one, it is frequently impossible to solve the equation for either of these variables, or to find y = fx or x = y. In cases of this kind we may suppose ƒ (x, y) = 0 to be a function of two variables which have a dependence on each other, and deduce the method of differentiation from Art. 54. Thus, let u = = f (x,y) But since u = dividing by dx, O, and y is a function of x, we have du = 0, and, du du dy we have only to divide the total differential (y2 - ax)3 = d'y d x2 (y2 — ax) (a3y — ax2 — 2xy2+2ax2) — (ay − x2) (2ay2 — 2x2y—ay2+a2x) 2xy (y-3axy + x') + 2 a3 x y (y2 - ax)3 = 2 a3xy |
