Derivation and integration of functions of a real variable

Description

Subject objectives: Understand and apply the fundamental concepts of the differentiation of real-valued functions of a single variable, including its main rules, properties, and associated theorems (Rolle’s theorem, the Mean Value Theorem, L’Hôpital’s Rule, etc.). Analyze the behavior of functions using higher-order derivatives, identifying relative extrema, inflection points, and producing graphical representations. Understand the concept of the definite integral based on its formal construction, and apply its properties for the effective computation of integrals. Relate differentiation and integration through the Fundamental Theorem of Calculus, and use techniques such as substitution and integration by parts to compute antiderivatives. Apply integral calculus to geometric and analytical problems, including the computation of areas, lengths, volumes, surfaces of revolution, and improper integrals. Contents: The course content is divided into two blocks: Block 1: Differentiation of single-variable functions Topic 1. Concept of derivative and properties (9h CLE) Concept of derivative. Chain rule and derivative of the inverse function. Derivatives of elementary functions. Rolle’s Theorem and the Mean Value Theorem. Monotonicity and differentiation. L’Hôpital’s Rule. Topic 2. Higher-order derivatives and properties (9h CLE) Relative extrema. Taylor polynomial. Remainder formulas. Characterization of relative extrema. Inflection points. Graphical representation of real-valued functions of one variable. Block 2: Integration of single-variable functions Topic 3. Definite integral and properties (9h CLE) Construction of the Riemann integral. Darboux sums. Riemann sums. Integrable functions. Properties of the integral. Topic 4. Relationship between differentiation and integration (9h CLE) Fundamental Theorem of Calculus. Change of variables theorem. Elementary antiderivatives. Integration by parts. Computation of antiderivatives. Topic 5. Applications of integral calculus (3h CLE) Computation of planar areas. Graph lengths. Volumes and surfaces of revolution. Topic 6. Improper integrals (3h CLE) Improper integrals. Convergence criteria.