values. Let A CB be a circle whose centre is Q and diameter A Q B; draw QC at right angles to AB; take any point Pin the circumference, and draw PM at right angles to A B. Now if we conceive the point P to move from B to C, the line P M will vary in magnitude from 0 to QC, the radius. Thus A M, BM, QM, and M P are all variable quantities, and the radius QC or QB is a constant quantity. Constant quantities are usually represented by the earlier letters, a, b, c, d; and those that are variable by the later ones, as u, v, x, y, z., In the expression a + bx + c x2, a, b, c, are constant is the variable. quantities, and 3. A function of a variable quantity is an expression involving that variable, and usually one or more constant quantities. Thus the expressions a x2+b, a sin x, a+a- log (b + x) are severally functions of the variable quantity x. (b+x) If the value of a quantity depend on that of another which is variable, the former is termed a function of the latter. Thus if log (b + x); u = a x2 + b, u = a sin x, u = a + a* · then, in each of these equations, u is a function of x. It is not to be understood that u is the same function of the variable x in all these equations, and as the term "function" may be denoted by a single letter, as f, p, 4, F, etc., the preceding equations may be written symbolically, thus u = Fx: u = fx u = фх where the different letters f, o, F denote different functions of the variable. The quantity is called the independent variable, and u the dependent variable. A quantity may be a function of several independent variables, as in the expression u = ax2 + bxy+cy2+mx+ny + p. Here u is a function of the independent variables x and y, and may be written u = f(x, y). Functions are distinguished into algebraic and transcendental. An algebraic function is one that may be expressed in a finite number of terms, and in which the variable is subjected to some of the elementary operations of algebra, as addition, subtraction, multiplication, division, involution and evolution. Thus, if m and n be finite, u = axTM + (b x2 - 2 cx) (a2 + x2)" is an algebraic function. A transcendental function is one that cannot be expressed in a finite number of terms, as u = log (1 − x), u = a", u = sin x. Functions are likewise either explicit or implicit. An explicit function is one in which the value of the dependent variable is exhibited in terms of the independent variable and constants, as u = a x2 + bx + c. A function is implicit when some operation is necessary for exhibiting its value in terms of the independent variable or variables. Thus in the equation u2 + 2 x u+a2 = 0, u is an implicit function of x. Resolving this equation for u, gives − x ±ˆ'√ (x2 − a2), and in this form u is an explicit function of x. 4. A variable is said to be continuous when in passing from one assigned magnitude to another it passes through all the intermediate ones, and a function of a quantity which varies continuously is said to be continuous between two assigned values, if in passing from one of them to the other it passes through all the intermediate values. Thus an angle may be made to vary continuously, increasing or diminishing by degrees, minutes, or seconds, or even the minutest fractions of seconds. A discontinuous variable, and a discontinuous function of a variable, do not fulfil the conditions of passing through all the intermediate values. Thus the tangent of an angle is continuous when the angle varies from O to 90; but it is discontinuous at 90, passing at once from a positive value to a negative one. In the differential and integral calculus all quantities are supposed to vary continuously, and to the student who is acquainted only with the ordinary processes of algebra, the principles of the calculus may, at first, appear somewhat unintelligible, especially as abstract quantities are considered as admitting of continuous change, and of taking certain finite ratios as they approach the limits of zero or of infinity. As the principle involved in a limiting ratio must be thoroughly understood, we shall first advert to some illustrations of it. ILLUSTRATIONS OF THE PRINCIPLE OF LIMITS. 5. Definition. The limiting value of an expression is the quantity towards which it continually approaches, by making the variable continually approach a certain value. Thus a circle is the limit of the area of an inscribed regular polygon. For by continually increasing the number of the sides of the polygon, its area will approach more and more to the area of the circle, and their difference may be made less than any quantity that can be assigned. The circle is consequently said to be the limit of the inscribed polygon when the number of sides approaches infinity. The following examples will tend to illustrate the important subject of limits. hence, as x approaches to a, h is successively diminished, and the limit = a + x, for all values of x, and therefore as x approaches to a, the more will a +x approach to 2 a. Thus, a a -x 3x+7 18, = 39, and so on. when x approaches. Dividing numerator and denominator by x, gives but as the value of x approaches ∞, the values of evanescence; therefore the limit required is 3 6 = 3+7 6 x 1 ia 2 x + a' ao 2 a x + x2 when x approaches 0. Ans. 2 a. 6. The elementary principles of the differential calculus may be illustrated by considering the changes produced on certain functions corresponding to changes in the variable.. Let u denote a function of the variable x, and u' the same function of x+h, that is, let u = ƒx, and u' = f (x + h), the latter function having the same form with respect to x+h which the former has with respect to x; then u' u will be the change produced on the function, corresponding to h, the change of the variable, and the relation between these simultaneous changes made on the function and the variable will be best exhibited by dividing u' u by h. Thus, if u = ax3, and u' = a (x + h)3 = a x2 + 3 a x2 h + 3 a x h2 + a h3; then will In both these instances the variable x receives the increment h, and the corresponding changes, or increments, of the functions are the expressions marked (1) and (l'). It will be observed that the first terms of the second members of (2) and (2') are independent of the increment h, and consequently if h be continually diminished down towards zero, the second members of (2) and (2) will tend to become simply their first terms, and by giving to h a sufficiently small value, each of them may be made to differ from its first term by a quantity less than anything that can be assigned. These first terms are the limits to which the second members, and consequently the first, tend when the increment h approaches more and more to zero. Thus in equation (2) the limit to which the quotient of u' u divided by h approaches, as his diminished towards zero, is 3 a x2; and though the terms of the fraction in the first member, viz., u' u and h suffer diminution by the continued diminution of h, yet the fraction itself does not suffer unlimited. diminution, but tends eventually to take as its value the first term, 3 a x2, of the second member of the equation. This will be obvious when we consider that the magnitude of a fraction does not depend on the actual magnitude of its terms, but on their comparative magnitudes. Thus 600 6 is the same as that of the fraction 700 7' the value of the fraction also the same as that of the fraction ⚫000000006 and Hence though, in the case of any proposed function, the increments of the variable and the function, h and u' u, may be made smaller and smaller, so as to become as nearly evanescent as we choose, it does not necessarily follow that the fraction arising from dividing u' u by h will become evanescent. It will be constant if the function be u= a x, and it will be diminishing towards a limit if the function be u = a x. Take the latter function, ua x3, and let a = 10, and x = 2. then by equation (2), the limit is 3 a x2 = 3 x 10 x 22 = 120. Now if and thus the smaller h is taken the more nearly will both members of equation (2) approach to the limit 120, the numerator as well as the denominator of the first member suffering rapid diminutions. Hence if u = a a 23, and if h be the increment of the variable x, then 3 a x2 h + 3 a x h2 + a h3 is the corresponding increment of the function u, is the ratio of the increment of the function to that of the variable, and the first term, 3 a x2, is the limit of the ratio of the increment of the function to that of the variable. 7. In general u may always be represented by the ordinate P M of some curve APQ, whose abscissa A M = x. Let RPT be a line touching the curve at P, and QPS a line cutting it in Q and P; then if the point Q be conceived to approach P, the line QPS will approach to coincidence with the line RPT. Draw QN parallel to P M, and PH parallel to the abscissa A M. Let h, then Q N = u' = ƒ (x + h), MN == A M N H But as Q approaches P, the intersection S will approach T, and there These illustrations will enable the student to form a correct notion of what is meant by a limit to the value of a variable function and a variable ratio. 8. It has been seen that if u = a x3, and u' = a (x + h)3, then This is termed the difference of the function u = u x3, and whenever the difference of a function can be expressed in terms containing successively h, ha, h3, etc., as multipliers, the first term of the difference is called the differential of the original function. Now if u x be the proposed function, then u' = x+h, and we have u' - u = h, so that h is the differential, as well as the difference of the function u = x. The differential of any quantity or function of any quantity, is indicated by prefixing to it the letter d, the initial of the word differential. Thus du is called the differential of u, dx the differential of x, and so on; but when u = x, then h is the differential of u or x; hence h = dx. Again, if u a ax, then u'a (x + h), and u' -u =2axh+ah2; hence, du or d (a x2) 2 axdx. The coefficient 2 ax, or the multiplier of dx is called the differential coefficient of the proposed function, and since in this case we have = it is obvious that the differential coefficient, 2 a x, is the limit of the ratio of the increment of the function to that of the variable. 9. From the preceding observations it is evident that the limit of the ratio of the simultaneous increments of a function, and of the variable on which it depends, will be different for different functions, and that there exists such a connexion between the function and the limit of the ratio that the one may be derived from the other. This connexion gives rise to an extensive and important analytical theory, consisting of the two following parts: I. A function of a variable quantity being given, to determine the limit of the ratio of the increment of the function to the increment of the variable, and conversely, II. Having given the limit of the ratio of the corresponding increments of a function and its variable, to determine the function. The former of these divisions is called the differential calculus, and the latter the integral calculus. The differential calculus is that branch of analysis whose object is to determine the limit of the ratio of the increment of any function to the increment of the variable, and to explain some of the principal uses which may be made of the limit of this ratio in pure mathematics. The differential calculus will consequently be founded on the following definition. If a variable quantity, x, be increased by a quantity h, and if any function of x be taken from the same function of x + h, and the remainder be divided by h, the limit to which the quotient so obtained will continually approach, and from which it may be made to differ by a quantity less than any that can be assigned, is called the differential |
