Functions of One Variable for Mathematics Entrance Exams

Mastering the calculus of functions of a single variable is crucial for succeeding in competitive mathematics exams. This comprehensive, text-only course offers a clear, rigorous pathway through real analysis and calculus, designed to build both conceptual clarity and exam-ready problem-solving skills. You will progress from foundational real-number properties to advanced theorems, learning how to dissect and solve complex exam problems. What you'll learn: Understand the fundamental properties of real numbers, bounds, and sequences; Master limits, continuity, and the concept of uniform continuity; Apply key differentiability concepts, including Rolle's Theorem and Mean Value Theorems; Evaluate definite integrals using Riemann integration and the Fundamental Theorem of Calculus; Solve advanced series expansion problems using Taylor's and Maclaurin's theorems; Practice rigorous mathematical proof techniques and analytical reasoning. The course begins with essential definitions of sets, intervals, and functions, ensuring your mathematical foundation is secure before advancing to limits, derivatives, and integration. It is designed for undergraduate mathematics students and exam candidates seeking a thorough, step-by-step written guide with no prior advanced calculus experience required. Start reading today to master the core concepts of single-variable calculus.