Calculus Set Free

Abstract

Abstract Calculus Set Free: Infinitesimals to the Rescue is a single-variable calculus textbook that incorporates the use of infinitesimals, and more generally the hyperreal numbers. The infinitesimal methods and notation herein were developed with beginning calculus students in mind, resulting in exposition that is more intuitive as well as many calculational procedures that are easier to perform, as compared to both traditional calculus textbooks and earlier attempts at including infinitesimals in calculus. Arithmetic of hyperreal numbers, levels of hyperreal numbers, and approximation in the hyperreals lead to a definition of limit. Limit computations are based directly on that definition. Computation-style proofs of derivative rules use an approximation formula called the “local linearity formula.” The definite integral is developed through the idea of finding area using infinitely many subintervals and right-hand endpoints; the resulting “omega sums” are much easier than Riemann sums as a result of the “sum of powers approximation formula,” which also anticipates the Fundamental Theorem of Calculus by its resemblance to the antiderivative power rule. The limit comparison test is replaced by the “level comparison test,” which is so widely applicable and computationally simple that strategy for testing series is noticeably less difficult. Although infinitesimal methods are used for any mathematical process involving a limit, the remainder of the text uses the standard methods of calculus. Organization is similar to other college-level calculus texts. Features include ample marginal notes, examples, illustrations, and answers to odd-numbered exercises.