Analysis I introduces students to the foundational concepts of real analysis, beginning with the properties of real numbers, including their completeness, order structure, and the notions of bounds, supremum, and infimum. The course covers sequences, focusing on their convergence, divergence, monotonicity, and boundedness, along with key results such as the Bolzano–Weierstrass theorem. Infinite series are introduced, with students learning how to determine convergence using tests like the comparison and ratio tests, and analyzing important examples such as geometric and harmonic series. Limits of functions are rigorously defined through the - approach, with exploration of their properties, computation rules, and notable limits. Continuity is studied both at a point and over intervals, supported by fundamental results like the Intermediate Value Theorem and properties of continuous functions on closed intervals. The course then develops the concept of differentiation, introducing the derivative as a limit, deriving rules of differentiation, and applying derivatives to study monotonicity and extremal points. Finally, students are introduced to Taylor and Maclaurin series as polynomial approximations of functions, learning key expansions such as those of , , , and , along with an understanding of the remainder term. This course lays the groundwork for deeper studies in mathematical analysis and problem-solving.
- معلم: Farida Derouiche