This course represents a bridge between high school and university. The main goal of the course is to provide the students with tools to follow the concatenation of simple logical arguments, and to introduce the fundamentals of differential and integral calculus for functions of one variable, with applications to ordinary differential equations of the first and the second order.

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.

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. Indefinite integral. The definite integral of a piecewise continuous function. 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 and trigonometric 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. Springer-Verlag UTX, 2015. Additional material will be provided by the teacher and will be available in the Internet personal page of the student.

Modalità di esame: Prova orale facoltativa; Prova scritta su carta con videosorveglianza dei docenti; Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;

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 on-going 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. They will be detailed by each lecturer and include, for example, answering to 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 with a proctoring system (Respondus), 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. Each question is worth one point, so that the maximum achievable score is 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 75 minutes, and typically consists of two exercises. Books or other teaching material cannot be consulted, as well as any electronic device not explicitly authorized. 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 on-going 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). ORAL EXAM. This part of the exam mainly aims at ascertaining the theoretical knowledge of the subject, and is a further element of appraisal. In case 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.

Modalità di esame: Test informatizzato in laboratorio; Prova scritta (in aula); Prova orale facoltativa; Prova scritta su carta con videosorveglianza dei docenti; Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;

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 on-going activities, in the test, in the written test, and and in the oral exam, if it takes place. ONGOING ACTIVITIES. The activities require the active participation of the student during the semester. They will be detailed by each lecturer and include, for example, answering to 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 with a proctoring system (Respondus), consists of 15 multiple-choice quizzes, including theoretical questions and covering all the syllabus. Books or other teaching material cannot be consulted, as well any electronic device not explicitly authorized. Each question is worth one point, so that the maximum achievable score is 15. If the score is less than 8 then the exam is failed, otherwise the student proceeds with the written exam. WRITTEN EXAM. This part lasts 75 minutes, and typically consists of two exercises. Books or other teaching material cannot be consulted, as well as any electronic device not explicitly authorized. The maximum achievable score is 15. If the score is less than 8 then the exam is failed, otherwise the final score of the exam is obtained as the sum of the scores of the on-going 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). ORAL EXAM. This part of the exam mainly aims at ascertaining the theoretical knowledge of the subject, and is a further element of appraisal. In case 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.