Advanced Calculus of a Single Variable
This advanced undergraduate textbook is based on a one-semester course on single variable calculus that the author has been teaching at San Diego State University for many years. The aim of this classroom-tested book is to deliver a rigorous discussion of the concepts and theorems that are dealt with informally in the first two semesters of a beginning calculus course. As such, students are expected to gain a deeper understanding of the fundamental concepts of calculus, such as limits (with an emphasis on ε-δ definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L’H˘pital’s rule) and the Riemann integral (discussing improper integrals in-depth, including the comparison and Dirichlet tests).
Success in this course is expected to prepare students for more advanced courses in real and complex analysis and this book will help to accomplish this. The first semester of advanced calculus can be followed by a rigorous course in multivariable calculus and an introductory real analysis course that treats the Lebesgue integral and metric spaces, with special emphasis on Banach and Hilbert spaces.
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ˇ ˇ ˇ antiderivative Assume that f beginning calculus bounded interval Cauchy condition Cauchy sequence comparison test continuous at x0 converges absolutely converges or diverges converges uniformly cos.x cos(x cosine defined Definition denote derivative dx converges Example f is continuous f is integrable f x0 function f Fundamental Theorem given series improper integral inequality infinite series kDmX kDnC1 least upper bound Leibniz notation Let f(x lim n!1 limit limn limy ln.x Mean Value Theorem monotone natural exponential function Œa;b open interval positive integer power series Proof Proposition radius of convergence rational numbers real numbers Riemann integrable secant line series 1X series converges absolutely series P 1nD1 series XL Show that f sin.x ſ║ f(x Solution Theorem of Calculus uniform convergence uniformly continuous