Calculus Foundations: Continuity, Differentiability, and Applications of Derivatives

Calculus is the mathematical language of change, yet many students struggle with the transition from rote memorization to deep conceptual understanding. This text-based course bridges that gap by breaking down complex mathematical theories into clear, digestible explanations. You will move from foundational definitions of limits to the practical mechanics of derivatives and their real-world applications. By working through structured written explanations and step-by-step mathematical proofs, you will build a rigorous intuition for how functions behave and change. You will learn to identify continuity, analyze points of non-differentiability, and apply derivative concepts to solve complex optimization and rates-of-change problems. What you'll learn: - Understand the formal mathematical definitions of limits, continuity, and differentiability - Analyze functions to identify points of discontinuity and non-differentiable sharp turns or cusps - Master the rules of differentiation, including the chain rule, implicit differentiation, and logarithmic differentiation - Apply derivatives to find rates of change, tangent lines, and normal lines - Utilize the Mean Value Theorem and Rolle's Theorem to analyze function behavior - Solve optimization and related rates problems using systematic mathematical frameworks This course begins with essential definitions and foundational theorems before moving into practical problem-solving techniques and application strategies. Each section is designed to reinforce your understanding through detailed, written step-by-step walkthroughs of classic calculus problems. This course is designed for high school students, early university students, or self-learners preparing for exams who want a clear, conceptual, text-based guide to calculus. No prior calculus knowledge is required, though a basic understanding of algebra and trigonometry is recommended. Start reading today to master the core principles of calculus and excel in your exams.