MATHEMATICS AND STATISTICS

The course aims to provide the basic knowledge of infinitesimal, differential and integral calculus of functions of one variable.

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.

Ability to argue and communicate, both orally and in written form. Knowing how to identify, describe, and work with sets. Recognizing hypotheses and theses of a theorem. Identifying whether a condition is necessary or sufficient. Knowing how to negate a proposition and understand a proof by contradiction. Understanding the difference between examples and counterexamples. Knowing numerical sets, particularly the algebraic and ordering properties of real numbers.

Knowing the definition, graph, and main properties of functions.

Being able to apply algebraic and monotonic properties of fundamental functions to solve simple equations and inequalities involving irrational, exponential, logarithmic, and trigonometric functions. Knowing the equations or inequalities of simple geometric figures (line, half-plane, circle, ellipse, hyperbola, parabola). Knowing the main trigonometric formulas.

1. BASIC CONCEPTS OF SET THEORY.

Set operations and properties. Functions. Domain, image, and graph of a function. Injective, surjective, and bijective functions. Infinite sets. Invertible functions. Composite functions. 

2. NUMERIC SETS.

Numeric sets N, Z, Q. Properties of rational numbers. The set of real numbers. Separated sets. Extremes of a numeric set.

3. LINEAR ALGEBRA.

Linear systems. Matrix associated with a linear system. Determinant.

4. PLANE GEOMETRY.

The Cartesian plane. Distance between two points. Midpoint of a segment. Symmetric point with respect to another point. Area of a triangle. Collinear points. Analytical representation of a line. Slope. Parallelism between lines. Trigonometric circle. Radian. Trigonometric functions and identities. Geometric meaning of the slope of a line. Perpendicularity between lines. Symmetric point with respect to a line. Distance from a point to a line. Parabola.

5. EQUATIONS AND INEQUALITIES.

Rational, fractional, irrational, logarithmic, exponential, and trigonometric equations and inequalities.

6. REAL FUNCTIONS OF A REAL VARIABLE.

Intervals. Neighborhoods. Accumulation points. Real functions of a real variable. Domain, image, and graph of a function. Supremum and infimum of a function. Monotonic, even, odd, and periodic functions. Elementary functions. Properties and qualitative graphs of elementary functions. Piecewise-defined functions. Domain search for real functions of a real variable.

7. LIMITS OF FUNCTIONS.

Definition of a limit. Limits of elementary functions. Lateral limits. Theorems of uniqueness of the limit, permanence of the sign, and comparison. Operations with limits. Indeterminate forms. Limits of monotonic functions. The number e. Notable limits. Asymptotes of the graph of a function. Definition and properties of continuous functions. Theorem of the existence of zeros* and intermediate values*. Weierstrass's theorem*. Continuity of monotonic functions*. Invertible functions. Continuity of inverse functions*.

8. DIFFERENTIAL CALCULUS.

Derivative of a function. Relationship between continuity and differentiability. Higher-order derivatives. Geometric meaning of the first derivative. Derivatives of elementary functions. Derivative of the sum, product, reciprocal, and quotient of functions. Differentiation of composite and inverse functions. Relative extremes. Theorems of Fermat, Rolle, and Lagrange and their consequences. Concavity, convexity, and inflection points. L'Hôpital's rules*. Graphs of elementary functions. Study of the graph of a function.

9. INDEFINITE INTEGRAL.

Antiderivatives. Indefinite integral. Immediate indefinite integrals. Properties of integrals. Integration methods by decomposition, by parts, and by substitution. Integration of rational fractional functions.

10. DEFINITE INTEGRAL.

Definite integral. Integral function. Mean value theorem. Fundamental theorem and formula of integral calculus. Geometric meaning of the definite integral. Rules for definite integration by parts and by substitution.

11. STATISTICS.

Population, sample, statistical unit, variables. Unidimensional frequency distribution. Histogram. Median. Boxplot. Arithmetic mean. Variance. Standard deviation. Joint distribution of two quantitative variables. Linear regression. Event. Random experiment. Frequentist conception of probability. Probability density function. Probability distribution. Normal probability distribution.

Topics marked with an asterisk are intended without proof.

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

8. M. Gionfriddo, Istituzioni di matematiche, Culc, Catania.

9. A. Guerraggio, Matematica per le scienze, Pearson.

10. V. Villani – G. Gentili, Matematica 5/ed Comprendere e interpretare fenomeni delle scienze della vita, Mc Graw-Hill.

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 15/30. The exam is considered passed if an oral interview is judged to be at least sufficient (18/30).

Definition of continuous function

Fermat theorem implications

Geometric definition of the concept of integral