B2288 - Principles of Mathematics T-1

Learning outcomes

At the end of the course the student knows the methodological-operational aspects of mathematical analysis and some of its applications, with particular regard to the functions of one variable.

Course contents

The course introduces the fundamental tools of mathematical analysis for the study of functions of a real variable. Particular emphasis is placed on mathematical modeling, the geometric interpretation of results, and the application of analytical techniques to the formulation and analysis of elementary problems arising in science and engineering. The course also provides the mathematical background required for the subsequent courses in mathematics of curriculum (Principles of Mathematics T-2 and Rational Mechanics T).

Course outline:

Basic concepts, functions, and mathematical modelling. Sets and logic. Number systems. Functions as mathematical models. Domain, codomain, and range. Graphs of functions. Monotonic functions. Injective, surjective, and bijective functions. Composition of functions. Inverse functions. Graph transformations. Elementary functions.

Linear systems. Systems of linear equations. Geometric interpretation in two variables. Matrix representation of a linear system. Gaussian elimination. Elementary discussion of the existence and uniqueness of solutions.

Sequences and limits. Sequences. Limits of sequences. Monotone sequences. Algebra of limits. Squeeze Theorem. Selected fundamental limits.

Limits and continuity of functions. Accumulation points. Limits of functions. One-sided limits. Algebra of limits. Limits of composite functions. Fundamental indeterminate forms. Continuity. Discontinuities. Asymptotes. Intermediate Value Theorem. Zero (Root Existence) Theorem.

Derivatives: definition and computation. Difference quotient. Derivative. Geometric and physical interpretation. Derivatives of elementary functions. Differentiation rules. Chain rule. Derivative of the inverse function. Points of non-differentiability.

Fundamental theorems on derivatives. Rolle's Theorem. Lagrange's Mean Value Theorem. Consequences of the Mean Value Theorem. Monotonicity. L'Hôpital's Rule.

Qualitative analysis of functions. Local and global maxima and minima. Weierstrass Theorem. Fermat's Theorem. Higher-order derivatives. Concavity. Inflection points. Qualitative analysis of functions. Geometric interpretation of the results. Taylor's Theorem.

Integration. Antiderivatives. Indefinite integrals. Integration by substitution. Integration by parts. Definite integrals. Properties of the definite integral. Mean Value Theorem for Integrals. Fundamental Theorem of Calculus. Area computation. Elementary geometric and physical applications.

Introduction to numerical methods. Numerical approximation. Absolute and relative error. Bisection method. Newton's method. Numerical integration.

Readings/Bibliography

The course materials consist of lecture notes, slides, and exercise sheets made available by the instructors throughout the course.

The main reference textbook is:

M. Bramanti, F. Confortola, S. Salsa, Matematica per le scienze. Con fondamenti di probabilità e statistica. Zanichelli, 2024.

The same textbook will also serve as the reference text for the course Principles of Mathematics T-2 in the 2027/28 academic year.

Teaching methods

The course consists of lectures and instructor-led classroom exercise sessions.

Assessment methods

The assessment consists of a written examination followed by an oral examination. Admission to the oral examination is conditional upon obtaining a minimum score of 16/33 in the written examination. The mark obtained in the written examination serves as the starting point for the oral examination. Following the oral examination, the final mark may be either higher or lower than the mark awarded for the written examination.

With regard to assessment, the use of artificial intelligence (AI) is prohibited (so-called Scenario 1). Any use of AI constitutes a violation of academic integrity.

Students with Specific Learning Disorders (SLD) or temporary or permanent disabilities: students are encouraged to contact the University's designated support office as early as possible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en). The office will propose any appropriate accommodations for eligible students. Any such accommodations must be submitted to the instructor for approval at least 15 days before the assessment. Approval will be granted at the instructor's discretion, taking into account the intended learning outcomes of the course.

Teaching tools

Tutoring sessions are provided to support problem-solving practice and exam preparation.

Office hours

See the website of Michele Ruggeri

See the website of Davide Palitta