Welcome to our intensive calculus course designed for completion in one month. We'll cover four key areas: limits and derivatives in week one, applications and basic integration in week two, advanced integration techniques in week three, and series plus multivariable calculus in week four. This structured approach requires daily commitment but will give you a solid foundation in calculus.
Week one focuses on limits and derivatives, the fundamental concepts of calculus. We start with understanding what a limit means - the value a function approaches as the input gets closer to a specific point. This leads us to continuity and then to derivatives, which represent the rate of change. You'll learn the power rule, product rule, quotient rule, and chain rule for finding derivatives.
Week two bridges derivatives and integrals through practical applications. You'll solve optimization problems like finding maximum profit or minimum cost, work with related rates where multiple quantities change simultaneously, and learn curve sketching techniques. The second half introduces integration as the reverse of differentiation, using the Fundamental Theorem of Calculus to find areas under curves.
Week three focuses on advanced integration techniques that handle complex functions. You'll master substitution for composite functions, integration by parts for products, and partial fractions for rational functions. These powerful methods, combined with trigonometric integrals and improper integrals, will enable you to solve sophisticated calculus problems that appear in physics, engineering, and advanced mathematics.
Week four introduces series and multivariable calculus. You'll learn convergence tests to determine if infinite series converge, explore Taylor series for function approximation, and begin multivariable calculus with partial derivatives and multiple integrals. Congratulations on completing this intensive month-long journey through calculus fundamentals! You now have the foundation for advanced mathematics, physics, and engineering applications.