2 - Limits, Derivatives and Series

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This chapter presents the definition of limit and its detailed procedure followed in the case of a function with example. The general rules for determining the derivatives of algebraic functions, as proven in all basic calculus courses, are summarized in the chapter. A number of expressions for the derivatives are derived from the problems at the end of the chapter. The operation of taking the derivative—that is, the result of the operator d/x operating on a function of x, followed by the same operation—yields the second derivative. The extrema of a function and its critical points is discussed with an example of a function, which exhibits an inflection point that is provided by the well-known Van der Waals equation. For practical purposes, the derivative dy/dx is decomposed into differentials in the form dy=(dy/dx)dx. The differential of a product of two functions is equal to the first function times the differential of the second plus the second times the differential of the first. Numerous examples of this principle are encountered in the exercises at the end of the chapter. The chapter presents the mean-value theorem and L'Hospital's rule, addresses power series to represent a function, and discusses the tests of series convergence as well as the most useful test for the convergence of a series called “Cauchy's ratio test.”