ANALISI MATEMATICA I

Module MODULO A

The lectures are complemented by exercises related to the topic of the course and both the lectures and the exercises will be carried out in frontal mode. It should also be noted that, for the Module B of the course, there are 35 hours of theory and 15 hours of other activities (typically, these are exercises). Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

Lecture attendance is not compulsory but it is strongly recommended.

1. Sets of numbers. 
• Natural numbers, integer numbers, rational numbers, real numbers. Basic notions on the set N of natural numbers, on the set Z of integer numbers and on the set Q of rational numbers. The set R of real numbers. Some consequences of the axioms on the real numbers*. Intervals. Absolute value of a real number. Bounds for numerical sets. The set N. Archimedean property*. Density of Q and R-Q in R *. Powers with real exponent*.
• Complex numbers. Basic definitions. Total order. Polar coordinates in the plane. Trigonometric form of a complex number. Product and power of complex numbers in trigonometric form. Exponential form of a comples number. Product and power of complex numbers in exponential form. nth roots of a complex number. Algebraic equations.
2. Functions and limits. 
• Functions. Basic definitions and properties.
  
• Limits. Distance and topology in R. Sequences: definitions, subsequences, limits, indeterminate forms, Napier's number, Cauchy sequences, Cauchy criterion for the convergence, Bolzano-Weierstrass Theorem. Limits of functions: sequential characterization of the limit of a function, theorems on limits, one-sided limits and theorem on limits of monotone functions, Theorem on the limit of the composite function.
  
3. Continuous functions and local comparison.
   
• Continuous functions. Definition of continuous function and basic results. Continuity of elementary functions and operations with continuous functions. Singularity points: removable singularity, singularity of the first kind and singularity of the second kind. Properties of continuous functions: local properties and global properties. Theorem on the existence of zeroes and its generalization, Intermediate value theorem, Weierstrass Theorem. Injectivity and strict monotonicity for continuous functions. Theorem on the continuity of the inverse function. Fundamental limits.
• Local comparison of functions. Bachmann-Landau symbols, comparison between infinitesimal and infinite functions. Asymptotes. Uniformly continuous functions and Lipschitz functions.
  
4. Differential Calculus.
• Definition of differentiable function and definition of derivative. Geometric and kinematic meaning of the first derivative. Relationship between continuity and differentiability. First finite increment formula. Derivatives of the elementary functions. One-sided derivatives. Point of non-differentiability. Definition of differential. Rules of differentiation. Theorem on the differentiability of the composite function. Theorem on the differentiability of the inverse function.
  
• Fundamental theorems of Differential Calculus and their consequences. Fermat's Theorem, Rolle's Theorem, Lagrange's Theorem and its consequences, De L'Hôpital's Theorem. Theorem on the limit of the derivative. Limit of the derivative and points of non-differentiability. Higher-order derivatives. Taylor formula with Peano's remainder and Taylor formula with Lagrange's remainder. Concave functions and convex functions and inflection points: definitions and theorems. Higher-order derivative test for the study of stationary points. Qualitative study of a function.
  

Recommended books for the Prerequisites

[P1] C.Y. Young, Algebra and Trigonometry. Fourth Edition, Wiley (2017).

[P2] C. Y. Young, Precalculus. Third Edition, Wiley (2018).

Recommended books for the course of Mathematical Analysis I

• Recommended books for the Theory:

[T2] R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).

• Recommended books for the Exercices:

[E2] R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).

During the period of delivery of the lessons, some self-assessment tests will be administered. These self-assessment tests have the task of guiding the student in the gradual learning of the contents displayed during the lessons. In addition, the self-assessment tests allow the teacher to quickly implement any additional activities aimed at supporting students in view of the exams.

Structure of the exam

The Mathematical Analysis I exam can be passed in two ways.

• Mode 1: mid-term written and oral tests

• Mode 2: one written test and one oral test (see Module B)

Mode 1: 

Mode 1 includes two intermediate tests, the first at the end of Module A, the second at the end of Module B: each intermediate test consists of a written test and an oral interview. 

Dates of the intermediate tests relating to Module A. 

There are three useful dates for the intermediate written test relating to Module A: two within the First Exam Session and one during the period of suspension of teaching activities scheduled for April. The dates of the midterm exam are available on the degree course website.

Structure of the intermediate written test relating to Module A.

Four exercises will be proposed in the intermediate written test relating to Module A and the duration of the intermediate written test is 120 minutes. 

Evaluation of the intermediate tests.

The maximum score that can be obtained in the written intermediate test is 30/30. The written intermediate test is considered passed if the student has achieved a score of at least 18/30. A score will be assigned to each exercise. The maximum score will be assigned if the exercise is carried out correctly, otherwise, a partial score will be assigned and it will be determined based on the errors made. If the total score is greater than or equal to 15 and less than 18, the Examination Commission may admit the student to the oral test with reservations if the student demonstrates adequate argumentation skills. 

Oral test on Module A. 

The oral exam for Module A must be taken within the Exam Session in which the written exam was passed. The student who does not pass the oral exam within the expected period will have the result of the written exam relating to Module A kept.

Note. Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of the Department of Electrical, Electronic and Computer Engineering.

All the topics mentioned in the program can be requested during the exam.

The attendance of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of ​​the questions that may be proposed during the exam.

An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the concepts exposed (definitions, propositions, theorems, corollaries).

The main types of exercises related to the Module A of the course of Mathematical Analysis I are:

• Finding the extrema of a numerical set.
• Exercises on complex numbers (algebraic manipulations, writing complex numbers in algebraic, trigonometric and exponential form, determination of the nth roots of a complex number, representations of geometric places in the Argand-Gauss plane, equations in the complex field).
• Calculation of limits.
  
• Study of the continuity and differentiability - and computation of derivatives - of real functions of a real variable.
• Classification of points of singularity and points of non-differentiability.
• Issues concerning the invertibility of functions and calculation of the derivative of inverse functions.
• Qualitative study of a function and applications (e.g., study of a function and its qualitative graph, proof of the existence of solutions of nonlinear equations).