Ability to follow a chain of logical arguments. Knowledge of the fundamentals of differential and integral calculus for functions of one variable. Gaining an adequate ability in computation.

Numerical sets, equations and inequalities, analytic geometry and trigonometry. Elementary functions and their basic properties. It is assumed that the prerequisites have been verified through the use of the material made available by Politecnico, already in the pre-enrollment phase.

Preliminaries: sets, operations with sets and logical symbols. Numerical sets, maxima, minima and extrema. The completeness property of real numbers and its consequences. Functions: surjectivity and injectivity; composition of functions, inverse functions. Functions of one real variable: elementary functions, monotone functions and inverse functions. (About 15 hours) Limits and continuity: limits of functions and sequences; continuity. Theorems on limits: uniqueness of the limit, sign-preserving property and local boundedness, comparison theorems. Limits of monotone functions. Algebra of limits. Indeterminate forms. Local comparison of functions. Landau symbols. Infinite and infinitesimal functions. Order of an infinity and of an infinitesimal, principal part with respect to a test function. Asymptotes. The number e. Fundamental trigonometric and exponential limits. Functions continuous over an interval: existence of zeros and of maxima and minima. (About 24 hours) Derivatives: geometrical and physical meaning. Computation of derivatives. Derivatives of elementary functions. Derivatives and continuity. Non-differentiability points, extremal and critical points. Fermat Theorem. Functions differentiable over an interval and fundamental theorems of differential calculus (Rolle and Lagrange) and their consequences. De L'Hôpital rule. Taylor formula and fundamental Maclaurin expansions. Use of Taylor expansions in the local analysis of functions: comparison, extrema and convexity. Applications to the study of the graph of a function. (About 23 hours) Primitives and their computation; primitives of rational functions. Definite integral. Properties of the integral. Integral mean, the mean value theorem and the Fundamental Theorem of Calculus. Relations between definite integrals and primitives. Improper integrals: definitions and convergence tests. (About 21 hours) Complex numbers and differential equations: algebraic, trigonometric and exponential form of a complex number. Real and imaginary part, modulus and argument. Roots of complex numbers; the Fundamental Theorem of Algebra. The exponential of a complex number and the Euler formula. Ordinary differential equations: the Cauchy problem. First order ordinary differential equations, linear or with separable variables. Second order linear differential equations with constant coefficients. (About 17 hours)

The course consists of 60 hours of lectures and 40 hours of exercise classes. Theoretical lessons are devoted to the presentation of the topics, with definitions, theorems, examples, properties and the proofs which are believed to facilitate the learning process. Every theoretical aspect is associated with introductory examples. The exercise hours aim to gain an adequate ability in computation.

C. Canuto, A. Tabacco. Mathematical Analysis I. Pearson, 2022. Additional material will be provided by the teacher and will be available in the Internet personal page of the student.

Text book; Exercises; Exercise with solutions ; Video lectures (previous years); Multimedia materials; Simulation tools; Self-assessment tools; Student collaboration tools;

You can take this exam before attending the course

Exam: Computer lab-based test; Written test; Optional oral exam;

Ongoing activities will be monitored, such as, for example, self-assessment tests and exercises during the entire duration of the semester. The main purpose of the exam is to ascertain both the theoretical knowledge of the subject and the ability to apply the theory to the solution of exercises. It consists of a multiple-choice test and a written exam, possibly followed by an oral exam. The final mark is determined taking into account the scores achieved in the ongoing activities, in the test, in the written test, and in the oral exam, if it takes place. ONGOING ACTIVITIES. The activities require the active participation of the student during the semester. Each lecturer will detail them and include, for example, answering self-assessment tests and the resolution of exercises to be delivered according to methods and deadlines announced at the beginning of the course. The maximum score is 3. TEST. This part of the exam, which lasts 45 minutes and takes place in a computer lab or a classroom with a proctoring system (LockDownBrowser), consists of 15 multiple-choice quizzes, including theoretical questions and covering all the syllabus. Books or other teaching material cannot be consulted, as well as any electronic device not explicitly authorized. A formula sheet will be made available. Each correct answer is worth one point, an incorrect or missing answer is worth zero points, and the maximum score is, therefore, equal to 15. If the score is less than 8, the exam is failed, otherwise, the student proceeds with the written exam. WRITTEN EXAM. This part lasts 90 minutes and typically consists of two exercises. Books or other teaching material cannot be consulted, as well as any electronic device not explicitly authorized. A formula sheet will be made available. The maximum achievable score is 15. If the score is less than 8, the exam is failed, otherwise, the final score of the exam is obtained as the sum of the scores of the ongoing activities, the test, and the written part, unless an oral examination is required by the lecturer (or by the student, provided that the final score is at least 18/30). Only for the winter exam session, a positive result in the test on the occasion of the first exam also grants access to the written part of the second exam (in particular, this applies in case of an insufficient result or refusal of the grade in the first exam). ORAL EXAM. This part of the exam mainly aims at ascertaining the theoretical knowledge of the subject and is a further element of appraisal. If this part of the exam takes place, its result will contribute (together with the scores of the other tests) to the final mark of the exam.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.