CHAPTER VI DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 91. Introduction. If y is a function of x, the average rate of change of y with respect to x in any interval Ax is Ay /Ax (page 35). If the average rate of change is constant, its value is the rate of change of y with respect to x (Definition, page 48), and the graph of y is a straight line (Theorem, page 50) whose slope is the rate of change. Numerous applications of uniform rate of change were given in Chapter II. If the average rate of change, Ay /Ax, is not constant, then the rate of change of y with respect to x is defined to be the limit of Ay/Ax as Ax approaches zero (page 94). It is represented graphically by the slope m of a line tangent to the graph of the function y. In this chapter we shall consider this limit more formally than heretofore, and derive rules for finding it expeditiously if y is an algebraic function of x. The applications are based either on the geometric interpretation of the limit of Ay /Ax as the slope of the tangent line or on the physical interpretation of the limit as the rate of change of y with respect to x. The ideas to be considered in this chapter, and the one following, are among the most fundamental and far-reaching concepts in mathematics. They were developed by the famous Englishman Sir Isaac Newton (1642-1727) and the noted German Gottfried Wilhelm von Liebnitz (1646-1716), and a bitter controversy lasting for many years was waged over the question as to which one of these men should be accorded the honor of the discovery. Leibnitz was the first to publish some of his results, in 1684, but Newton had written a paper on the subject and submitted it to some of his friends in 1669. The followers of each claimed that the other had been guilty of claiming ideas not his own, but most historians of mathematics are agreed that the work of Newton and Leibnitz was independent. On the basis of the work of these men, there followed a period of rapid and extensive mathematical development. Let us first consider the underlying concept of the limit of a variable. 92. Limits. By definition (footnote, page 93) the limit of a variable x is a constant such that the numerical value of the difference between x and a becomes and remains as small as we please. This explains what is meant by saying that "x approaches a as a limit," or in more compact form, "x approaches a." It is immaterial whether or not x becomes equal to a. A function of x is a second variable y (Definition, page 5). The functions we have studied are such that if x approaches a limit, so also does y, provided that y does not become infinite as x approaches a. The notation is used to indicate that "the limit of y, as x approaches a, is b." This means that the numerical value of the difference between y and b can be made as small as we please by taking x sufficiently near to a. Graphically, the difference between the ordinates y and b can be made as small as we please by making the difference between the abscissas x and a sufficiently small. In other words, Ay by approaches zero if Axa x approaches zero. = For example, the average rate of Y Ду P Ax b α FIG. 157. 80 change of the function ax2 in an interval Ax beginning at a definite point x is (see (5), page 95) The rate of change of y with respect to x for a given value of x is therefore m = lim Ay We are thus led to find the limit of 2ax + a Ax, a function of Ax, as Ax approaches zero. In computing this limit, x has a given value and is regarded as constant. In order to prove the assumption made earlier that the limit is 2ax, it must be shown that a Ax, the difference between the variable 2ax + a Ax and the constant 2ax, can be made as small as we please by making Ax sufficiently small. This follows readily. For if we wish to make a Ax as small as 0.001, it is sufficient to take Ax = 0.001/a, which is possible since Ax approaches zero and can therefore be made as small as we please. The limits encountered in computing lim Ax÷0 Ay Ax' where y y is a polynomial or a rational function of x, can be computed by means of the following theorems, which we assume without proof. Theorem 1. The limit of the sum of several variables is the sum of their limits. Theorem 2. The limit of the product of several variables is the product of their limits. Theorem 3. The limit of the quotient of two variables is the quotient of their limits, provided the limit of the divisor is not zero. The limit of the difference of two variables may be found by Theorem 1, since u V: = u + (− v). In applying these theorems, it is frequently convenient to regard y = c, a constant, as a function of x, whose limit, as x approaches a, is c. Give the details of the computation of the limit of Ay/ Ax as Ax approaches zero. lim 3x2 + lim 3x Ax + lim Ax Ax (Theorem 1) Ax÷0 Ax÷0 Ax÷0 Ax lim Ax (Theorem 2) 93. Derivative of a Function. DEFINITION. If y is a function of x, and if Ay is the change in y corresponding to a change of Ax in x, then the limit of Ay/Ax, as Ax approaches zero, is called the derivative of y with respect to x. Denoting it by Day (read "the derivative of y with respect to x"), the definition may be expressed by the equation The results obtained in Section 33, page 94, may now be stated as follows: The derivative of y with respect to x, Day, measures the rate of change of y with respect to x. sive steps in the process have been given in the section cited above. -x+Ax FIG. 158. by Theorem 3 of the preceding section. Applying Theorem 1 in the numerator and Theorems 1 and 2 in the denominator, we get 1. Evaluate the following limits, indicating in detail the use of the theorems in Section 92. (g) y = √x. Hint. Rationalize the numerator of Ay/Ax before passing to the limit as Ax approaches zero. 3. If f(x) = (x2 + 3x)/(x2 − 1), find lim f(x). Note that the value obtained is f(a), provided a ±1. What happens if a = ± 1? 4. If f(x) = x2, prove the relations 5. Prove that the relations in Exercise 4 are true if f(x) is any quadratic function. |