Summary Calculus I Foundations: Functions, Limits & Differentiation

This document is the first half of a comprehensive university-level Calculus I study guide covering the foundational concepts needed to succeed in calculus. These handwritten notes are detailed, organized, and designed to simplify difficult topics through explanations, examples, diagrams, formulas, and problem-solving methods commonly taught in college calculus courses. Part 1 focuses heavily on the conceptual foundation of calculus, beginning with functions and limits and progressing into derivatives, differentiation techniques, logarithms, exponential functions, and inverse functions. Topics Covered: • Functions & Graphs Function notation and evaluation Piecewise functions Even and odd functions Polynomial, rational, root, and logarithmic functions Domains and ranges Transformations and graph behavior Parent functions and graph recognition Composition of functions • Limits & Continuity Limit laws and substitution methods One-sided limits Infinite limits and limits at infinity Vertical and horizontal asymptotes Squeeze Theorem Continuity and discontinuities Intermediate Value Theorem Evaluating complex limits step-by-step • Introduction to Derivatives Definition of the derivative Difference quotient Tangent lines and slope interpretation Instantaneous rate of change Velocity and motion interpretation Differentiability and continuity • Differentiation Rules Product rule Quotient rule Chain rule Implicit differentiation Higher-order derivatives Related rates concepts Linear approximation and differentials • Exponential, Logarithmic & Inverse Functions Exponential growth and decay Properties of logarithms Derivatives of logarithmic and exponential functions Inverse functions and inverse trigonometric functions Logarithmic differentiation Change of base formula These notes combine lecture material, worked examples, visual sketches, highlighted reminders, shortcuts, and conceptual explanations in a way that is easier to understand than many textbooks. The notes are especially useful for students preparing for exams, quizzes, homework assignments, or cumulative finals. The material is written in a student-friendly format that emphasizes understanding concepts rather than memorizing formulas, making it ideal for self-study and review.