ID:
500115
Duration (hours):
84
CFU:
9
SSD:
ANALISI MATEMATICA
Located in:
PAVIA
Year:
2025
Overview
Date/time interval
Primo Semestre (25/09/2025 - 14/01/2026)
Syllabus
Course Objectives
The course aims to provide basic knowledge of differential and integral calculus for real-valued functions of a real variable, as well as techniques for the study of numerical sequences and series. The expected learning outcomes include: understanding of the fundamental concepts of Mathematical Analysis (for functions of one variable), mastery of proof techniques, development of computational skills and logical-deductive reasoning abilities, acquisition of a specific and appropriate scientific vocabulary, and adoption of a rigorous methodological approach.
Course Prerequisites
Basic knowledge of elementary algebra, trigonometry, and analytic geometry in the plane. Algebraic equations and inequalities. Elementary functions.
Teaching Methods
Lectures given by the teachers of the course. A significant part of the lectures will be devoted to examples and exercises. Attending the lectures is strongly recommended.
Assessment Methods
The exam consists of a written test and an oral exam. In the written test students will be asked to solve some exercises in differential and integral calculus and to answer some theoretical questions. The exam time for the written test is at most 3 hours. Its purpose is to assess both the understanding of the main theoretical results and the ability to apply the tools introduced during the course to explicit problems. Students can take the oral exam only if they score at least 15/30 on the written test. The results of the written test will be communicated by email. The oral exam consists of several questions aimed at assessing the student's understanding of the entire course content and their communication skills. The questions will vary in difficulty to determine the depth of the acquired knowledge. The final grade will be based on the combined results of the two parts of the exam.
Texts
E. Acerbi, G. Buttazzo: Primo corso di Analisi Matematica. Universitas Editore Parma, 2023. Further references: E. Giusti: Analisi Matematica 1. Bollati Boringhieri, 2002.
Contents
Axiomatic definition of real numbers: algebraic, order, and completeness axioms. Upper bound, lower bound, maximum, minimum, supremum, and infimum. Cardinality: finite, infinite, countable sets. Cardinality of N, Z, Q, and R. Archimedean property. Density of Q in R. Basic notions of topology: interior points, isolated points, and accumulation points.
Injective, surjective, bounded, monotone, convex functions. Even, odd, periodic functions. Elementary functions and their inverses.
Definition of limit for a sequence. Uniqueness of the limit theorem. Existence of the limit for monotone sequences theorem. Permanence of sign theorem. Comparison theorem. Sandwich theorem. Algebra of limits and indeterminate forms. Subsequences. Bolzano-Weierstrass theorem. Standard limits. Ratio test. Asymptotic relation. Liminf and limsup of a sequence.
Definition of series. Necessary condition for convergence. Geometric series, generalized harmonic series, telescoping series. Convergence tests for positive-term series: comparison test, limit comparison test, ratio test, root test. Cauchy condensation test. Leibniz test. Absolute convergence. Absolute convergence test.
Definitions of limit for functions. Uniqueness of the limit theorem. Existence of the limit for monotone functions theorem. Permanence of sign theorem. Comparison theorem. Sandwich theorem. Algebra of limits and indeterminate forms. Orders of infinity and infinitesimal. Definition of continuous function. Intermediate Value Theorem. Weierstrass Theorem. Continuity and monotonicity. Uniform continuity. Heine-Cantor theorem.
Definition of derivative and its physical and geometric interpretation. Derivative of sum, product, quotient. Derivative of composition. Derivative of inverse function. Tangent line. Differentiability. Continuity theorem for differentiable functions. Fermat theorem. Rolle, Lagrange, and Cauchy theorems. Absolute and relative maximisers and minimisers. Critical points. Monotonicity criterion. Maximisation and minimisation problems. L'Hôpital theorem. Taylor polynomials: definition and properties. Peano and Lagrange remainders. Taylor expansions for elementary functions. Convexity criteria. Characterization of monotonicity and convexity using first and second derivatives. Inflection points.
Definition of Riemann integral. Cauchy sums. Linearity of integrals. Monotonicity of integrals. Integrability of continuous functions. Mean value theorem for integrals. Integral domain splitting theorem. Primitives. Integral function. First and second fundamental theorem of integration. Integration by parts and by substitution.
Injective, surjective, bounded, monotone, convex functions. Even, odd, periodic functions. Elementary functions and their inverses.
Definition of limit for a sequence. Uniqueness of the limit theorem. Existence of the limit for monotone sequences theorem. Permanence of sign theorem. Comparison theorem. Sandwich theorem. Algebra of limits and indeterminate forms. Subsequences. Bolzano-Weierstrass theorem. Standard limits. Ratio test. Asymptotic relation. Liminf and limsup of a sequence.
Definition of series. Necessary condition for convergence. Geometric series, generalized harmonic series, telescoping series. Convergence tests for positive-term series: comparison test, limit comparison test, ratio test, root test. Cauchy condensation test. Leibniz test. Absolute convergence. Absolute convergence test.
Definitions of limit for functions. Uniqueness of the limit theorem. Existence of the limit for monotone functions theorem. Permanence of sign theorem. Comparison theorem. Sandwich theorem. Algebra of limits and indeterminate forms. Orders of infinity and infinitesimal. Definition of continuous function. Intermediate Value Theorem. Weierstrass Theorem. Continuity and monotonicity. Uniform continuity. Heine-Cantor theorem.
Definition of derivative and its physical and geometric interpretation. Derivative of sum, product, quotient. Derivative of composition. Derivative of inverse function. Tangent line. Differentiability. Continuity theorem for differentiable functions. Fermat theorem. Rolle, Lagrange, and Cauchy theorems. Absolute and relative maximisers and minimisers. Critical points. Monotonicity criterion. Maximisation and minimisation problems. L'Hôpital theorem. Taylor polynomials: definition and properties. Peano and Lagrange remainders. Taylor expansions for elementary functions. Convexity criteria. Characterization of monotonicity and convexity using first and second derivatives. Inflection points.
Definition of Riemann integral. Cauchy sums. Linearity of integrals. Monotonicity of integrals. Integrability of continuous functions. Mean value theorem for integrals. Integral domain splitting theorem. Primitives. Integral function. First and second fundamental theorem of integration. Integration by parts and by substitution.
Course Language
Italian
More information
Students belonging to one of the groups identified by the innovative teaching project may request to have access to the teacher's lecture notes and to schedule office hours also in an online format and by appointment, at times to be agreed upon with the teacher.
Degrees
Degrees
MATHEMATICS
Bachelor’s Degree
3 years
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