72
Mathematical Analysis
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
Aim of the course is to provide the basic concepts of differential and integral calculus of real functions, of numerical sequences and series and to present the field of complex numbers in order to analyze, set up correctly, with the necessary logical rigor, a mathematical problem.
Expected learning outcomes (according to Dublin Descriptors)
Knowledge and understanding: Upon passing the exam, the student will know the fundamental principles of infinitesimal, differential, and integral calculus. The student is familiar with operations in the complex field and with numerical series.
Applying knowledge and understanding: Upon passing the exam, the student will be able to use the tools of infinitesimal, differential, and integral calculus, as well as complex numbers and numerical series, in order to formalize and solve problems related to the structural disciplines of the course of study.
Making judgements: To pass the exam, the student must be able to recognize the most basic techniques of mathematical analysis and identify situations and problems where these techniques can be applied.
Communication skills: To pass the exam, the student must be able to articulate the theoretical motivations underlying the chosen calculation procedure for solving an exercise, using appropriate scientific language, as well as the logical reasoning behind the fundamental theorems of Mathematical Analysis.
Lifelong learning skills: Upon passing the exam, the student will be capable of independently deepening their acquired knowledge and applying it to understand new topics where mathematical analysis is applied.
Course Prerequisites
Syllabus required by TOLC-I
Teaching Methods
The course is organized according to the following breakdown:
- Lectures (42 hours): Explanation of definitions, theorems, and proofs.
- Exercise Classes (30 hours): Practical applications, function analysis, calculation of limits and integrals, series, and complex numbers.
Assessment Methods
The examination consists of two parts: a written exam and an oral exam.
The written exam lasts two hours and thirty minutes. It is designed to assess the student's ability to solve exercises regarding infinitesimal, differential, and integral calculus, complex numbers, and numerical series. Passing the written exam is a prerequisite for admission to the oral exam.
The oral exam is designed to assess the student's level of knowledge and understanding of the course content, as well as to evaluate independent judgment, learning skills, and communication abilities. The oral exam consists of a discussion of the written test, along with questions on the topics covered during the course.
The final grade is determined by taking into account the performance in both the written and oral components.
Evaluation Criteria
- 30 with honors (Lode): Complete, in-depth, and critical knowledge of the topics; excellent command of technical language; thorough and original interpretative skills; full ability to independently apply knowledge to solve proposed problems.
- 28 – 30: Complete and in-depth knowledge of the topics; great command of technical language; thorough and effective interpretative skills; ability to independently apply knowledge to solve proposed problems.
- 24 – 27: Knowledge of the topics with a good degree of mastery; good command of technical language; correct and confident interpretative skills; good ability to correctly apply most of the knowledge to solve proposed problems.
- 20 – 23: Adequate knowledge of the topics but limited mastery; satisfactory command of technical language; correct interpretative skills; more than sufficient ability to independently apply knowledge to solve proposed problems.
- 18 – 19: Basic knowledge of the main topics; basic knowledge of technical language; sufficient interpretative skills; sufficient ability to apply the basic knowledge acquired.
- < 18 (Fail): The student does not possess an acceptable knowledge of the topics covered during the course.
Texts
R. Adams, C. Essex, Calculus: A Complete Course, Seventh Edition (7th Edition)
Contents
Real numbers and functions. Basic notions. Sets. Elements of mathematical logic. Set of numbers: natural, integer, rational and real numbers. Principle of Mathematical Induction. Definition of function. Injective, surjective and bijective functions. Inverse function. Composition of functions. Elementary functions. Domain. Basic notions of topology. Bounded sets. Supremum and infimum of a set. Bounded functions. Supremum and infimum of a function. (1.5 CFU)
Complex numbers. Set of complex numbers. Algebraic operations. Cartesian coordinates. Trigonometric and exponential form. Powers and nth roots. (0.5 CFU)
Continuity. Definition of limit. Algebra of limits. Indeterminate forms. Theorems on limit: uniqueness and sign of the limit, comparison theorem. More fundamental limits. Asymptotes. Infinitesimal and infinite functions. Substitution theorem. Definition of continuous function. Points of discontinuity. Properties of continuous maps: Extreme Value Theorem; Intermediate Value Theorem. Corollary. Continuity of inverse functions. Continuity of composite functions. (2 CFU)
Differential calculus. Definition of derivative and geometric interpretation. Derivative and operations. Derivative and continuity. Derivative of composite functions. Derivative of inverse functions and applications. Higher-order derivatives. Extrema and critical points. Fermat Theorem. Theorems of Rolle, Lagrange, Cauchy. Geometric interpretation and consequences of Lagrange Theorem. Theorem of De l'Hôpital. Convexity and inflection points. Qualitative study of a function. Taylor and Mac Laurin formulas. Remainder term. Lagrange form of the remainder term. Application to calculus of limits. Peano form of the remainder. Application to calculus of approximation error. Hyperbolic functions. (2.5 CFU)
Integral calculus. Partition of an interval. Riemann sums. Riemann integral. Definite integrals. Integrability conditions. Properties of definite integrals and geometric interpretation. Integral mean value. The Fundamental Theorem of integral calculus. Primitive functions and indefinite integrals. Integrating by parts and by substitution. Integrating rational maps. Computation of areas. Improper integrals, Integrability conditions. Absolute convergence of integrals. (2 CFU)
Numerical sequences and series. Real sequence. Limit of a sequence. Limits of monotone sequences. More fundamental limits. Subsequences. Properties. Convergent, divergent numerical series. Geometric series, Mengoli series, harmonic series. Positive-term series. Comparison, Root, and Ratio Tests. Absolutely convergent series. Alternating series. Leibnitz Theorem. (0.5 CFU)
More information
Team Code: 5ynt5m7