ANALISI MATEMATICA I

The course aims to provide fundamental knowledge of differential and integral calculus for functions of a single variable. This foundational understanding will be applicable across various fields that utilize mathematical methods, such as sustainable energy exploitation, infrastructure design and atmospheric modeling to mitigate climate change effects. These applications align with Goals 7, 9, 11, 12, and 13 of the 2030 Agenda.

Knowledge and understanding:
This Mathematical Analysis I course tailored for Engineering students aims to provide a strong foundation in the essential mathematical concepts of real numbers, continuous functions, derivatives, integrals, and series, enabling students to apply these tools effectively in engineering problem-solving. Through rigorous study and practical applications, students will develop an appropriate knowledge and understanding of these mathematical principles, equipping them with the analytical skills necessary for success in their engineering coursework and future career.

Apply knowledge and understanding:
Students are encouraged to leverage their grasp of the mathematical tools to solve engineering problems. Through practical exercises and real-world applications, students will develop the ability to employ these mathematical concepts as powerful tools in engineering analysis and design.

Expressing judgments:
Students will be challenged to express informed judgments by evaluating the appropriateness and accuracy of mathematical techniques when applied to engineering problems. They will develop the ability to critically assess and select the most suitable mathematical methods, enhancing their problem-solving skills and engineering decision-making processes.

Communication skills:
The course places a strong emphasis on developing effective communication skills, equipping students with the ability to articulate mathematical concepts and problem-solving approaches clearly and concisely. Through exercises and collaborative discussions, students will learn to convey complex mathematical ideas to both technical and non-technical audiences, a crucial skill for success in their engineering careers.

Learning skills:

Students will actively cultivate essential learning skills, including self-directed study, problem-solving strategies, and adaptability when approaching mathematical challenges. Through a variety of exercises and assessments, students will develop the ability to independently explore and apply mathematical concepts, fostering a lifelong capacity for continued learning in engineering and related disciplines.

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 our Department, prof. Maurizio Spina.

Ability to communicate, orally and in writing. Knowing how to identify, describe and operate with sets. Recognize hypotheses and thesis of a theorem. Recognize whether a condition is necessary or sufficient. Knowing how to deny a proposition and understand reasoning by contradiction. Understand the difference between examples and counterexamples. Knowing numerical sets and, in particular, the algebraic properties of real numbers.

Knowing the definition, graph and main properties of functions.

Knowing how to apply the algebraic properties of the fundamental functions for the solution of simple irrational, exponential, logarithmic and trigonometric equations and inequalities. Knowing the equations or inequalities of simple geometric objects (line, semi-plane, circumference, circle, ellipse, hyperbola, parabola). Knowing the main trigonometric formulas.

1. OUTLINE OF SET THEORY
Set operations and properties. Functions. Domain, image and graph of a function. Injective, surjective and bijective functions. Infinite sets. Invertible functions.  Compound functions. Binary relationships. Equivalence and order relation. Ordered sets.

2. NUMERICAL SETS
The sets N, Z, Q. Properties of rational numbers. The set of real numbers. Separate sets.  Extremes of a numerical set. Power with natural and integer exponent. Existence and uniqueness of the nth root. Solvability of the equation x^n=a. Power with rational and real exponent. Logarithms. Absolute value. Rational, fractional, irrational, with absolute value, logarithmic, exponential and trigonometric equations and inequalities. Induction principle.

3. REAL FUNCTIONS OF A REAL VARIABLE
Intervals. Accumulation points. Real functions of real variables. Domain, image and graph of a function. Upper and lower bound of a function. Monotonic, even, odd, periodic functions. Elementary functions. Properties and qualitative graphs of elementary functions. Piecewise functions. Domain of real functions of real variable.

4. LIMITS OF FUNCTIONS AND SEQUENCES
Definition of limit. Limits of elementary functions. Limit of sequences.  Theorems of uniqueness of the limit, sign permanence theorem and comparison theorem. Limit operations. Indeterminate forms. Bounded sequences. Extremes of a sequence.  Limit of monotonic functions. Monotone sequences. Neper number. Limit of composite functions.  Link between limits of functions and of sequences. Notable limits. Sub-sequences. Asymptotes to the graph of a function.

5. CONTINUOUS FUNCTIONS
Definition and properties of continuous functions. Theorem of existence of zeros and intermediate values. Image of a continuous function in an interval. Weierstrass theorem. Continuity of monotonic functions. Invertible functions. Continuity of inverse functions.

6. SERIES
Character of a series. Mengoli series, geometric, harmonic series. Telescopic series. Necessary condition for the convergence of a series. Operations with series. Series with non-negative terms. Comparison, ratio and root tests. Generalized harmonic series.  Absolutely convergent series. Series with alternating signs. Leibniz criterion.

7. DIFFERENTIAL CALCULUS
Derivative of a function. Relationship between continuity and differentiability. High-order derivatives. Geometric meaning of the first derivative. Derivatives of elementary functions. Derivative of the sum, product, reciprocal and quotient functions. Derivative of composite functions and inverse functions. Relative extremes. Fermat, Rolle and Lagrange theorems and its applications. Concavity, convexity and inflections. De L'Hospital's theorems. Graphs of elementary functions. Study of the graph of a function. Taylor expansion.

8. INDEFINITE INTEGRAL
Primitive. Indefinite integral. Trivial indefinite integrals. Integral properties. Methods of integration by decomposition, by parts and by substitution. Integration of fractional rational functions.

9. DEFINITE INTEGRAL
Riemann integral. Condition of integrability. Classes of integrable functions. Properties of the Riemann integral. Mean theorem. Fundamental theorem and formulas of integral calculus. Geometric meaning of the definite integral. Rules of integration defined by parts and by substitution. Generalized and improper integrals.

REFERENCE BOOK LIST

1. M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, Mc Graw Hill 

2. G. Fiorito, Analisi Matematica 1, Spazio Libri

3. P. Marcellini, C. Sbordone, Analisi Matematica 1, Liguori

4. C.D. Pagani, S. Salsa, Analisi Matematica 1, Zanichelli

5. M. Bramanti, Esercitazioni di Analisi Matematica 1, Esculapio

6. T. Caponetto, G. Catania, Esercizi di analisi Matematica 1, Culc.

7. P. Marcellini, C. Sbordone, Esercitazioni di Matematica, Vol.1, Parte I e II, Liguori

The final exam consists of a written test and an oral interview. Access to the oral interview is granted if the written test is passed with a grade of no less than 16/30. The exam is considered passed if an oral interview is judged to be at least sufficient (18/30).

Booking for an exam session is mandatory and must be made exclusively via the internet through the student portal within the set period.

Criteria for assigning marks: both for the written and the oral exams, the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that they have acquired sufficient knowledge of the main topics covered during the course and that they are able to carry out at least the simplest of the assigned exercises.

The following criteria will normally be followed to assign the grade:

Not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.

18-23: the student demonstrates minimal mastery of the basic concepts, their skills in exposition and connection of contents are modest, they are able to solve simple exercises.

24-27: the student demonstrates good mastery of the course contents, their presentation and content connection skills are good, they solve the exercises with few errors.

28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; they solve the exercises completely and without mistakes.

Definition of continuous function,

proof of the zero theorem,

geometric definition of the concept of integral.