Free online courseCalculus I entire course

Course Description

Embark on a comprehensive journey through the essential concepts of calculus with "Calculus I," an extensive course that spans 16 hours and 1 minute. Tailored for those venturing into the core principles of calculus, this course is nestled within the Basic Studies category and falls specifically under Calculus. As of now, it awaits its first review, presenting an uncharted opportunity for students to gain a fresh perspective on the intricate world of calculus.

The course begins with a preliminary review of foundational topics such as factoring, graphing, and understanding linear models and rates of change. These initial lessons lay the groundwork for delving into the heart of calculus, ensuring that students are well-prepared to tackle more complex concepts. A significant emphasis is placed on functions and their graphs, alongside a detailed review of trigonometric functions and how to solve trigonometric equations.

Advancing into the crux of calculus, students are introduced to the concept of limits, which serve as the cornerstone of calculus. Through numerical and graphical approaches, the course elucidates how to determine limits, including those that do not exist, and delves into the epsilon-delta definition. Properties of limits and techniques for finding limits of indeterminate form functions are thoroughly explored, ensuring students grasp the underpinnings of continuity and its properties. The Intermediate Value Theorem, infinite limits, vertical asymptotes, and the fundamental principles surrounding them are analyzed in detail.

The course progresses to the study of derivatives, beginning with an introduction to the derivative using the slope of the tangent line and the definition of a derivative. Basic differentiation rules are rigorously covered, followed by practical applications of derivatives, such as the product and quotient rules, trigonometric and higher-order derivatives, and the essential Chain Rule. Differentiation strategies, implicit differentiation, and real-world applications in related rates are thoroughly discussed, providing students with a solid understanding of how derivatives function in various contexts.

Students then explore the critical concepts of extrema, Rolle's Theorem, and the Mean Value Theorem, which are pivotal for understanding the behavior of functions. The first and second derivative tests, along with techniques for determining intervals of concavity and points of inflection, are explained in depth. Detailed exploration of curve sketching, using both graph attributes and derivatives, prepares students to tackle optimization problems and understand tangent line approximation, differentials, and propagated errors.

The final section of the course introduces antiderivatives and differential equations, including sigma notation, summation formulas, and approximating the area under curves. The course meticulously covers Riemann Sums, definite integrals, and the Fundamental Theorems of Calculus, ensuring a holistic understanding of integration techniques, including integration by substitution.

Concluding with logarithmic, exponential, and inverse functions, the course provides a robust review of their properties, differentiation, and integration techniques, including the challenging applications of L'Hopital's Rule in resolving indeterminate forms.

Designed to provide a deep, foundational understanding of calculus, "Calculus I" is the optimal choice for students seeking to master this critical mathematical discipline. This course's meticulously structured content bridges essential pre-calculus concepts with the sophisticated intricacies of calculus, paving the way for academic and professional success in mathematics and related fields.