07166 - Fundamentals of Mathematics I

Learning outcomes

Upon completion of this course, the student knows the basic tools of mathematical analysis and linear algebra. He/she is able to use mathematical tools for the study of other disciplines.

Course contents

Fundamental concepts and tools. Sets, numerical sets, extremal points: supremum and infimum.

Functions. Definition of functions, domain, range, graphic; monotone functions, injective, surjective, bijective functions; function composition and inverse of a function. Elementary functions of real numbers.

Limits of functions of real variable. Cluster points. Definition of limit. Right and left limits. Asymptotes of functions. Computation of limits: fundamental limits, operations, limits of composition of functions. Continuous functions and points of discontinuity. Continuous functions on an interval: Bolzano’s theorem, intermediate values, Weierstrass.

Derivatives of functions of real variable. Derivative of a function at a point: analytical and geometric interpretations. Derivative as a function and its computation: derivative of elementary functions, derivation rules. Differentiable functions and point of non-differentiability. Lagrange’s mean value theorem and its conseguences, monotonicity. Relative maxima and minimia and the Fermat’s theorem. De L’Hopital’s theorem. Higher-order derivatives: inflection points and concavity. Overview on Taylor’s series.

Anti-derivation and integrals. Indefinite integral and the inverse of derivation. Computation of indefinite integrals: integration by parts and by substitutions, and some rational functions. Definite integral of continuous real functions: properties and integral mean theorem. Fundamental theorem of calculus. Some applications in physics and geometry.

Complex numbers. Imaginary unit, definition and operations between complex numbers. Polar representation: powers and roots of complex numbers. Fundamental theorem of algebra (statement).

Linear Algebra. Matrices and their operations. Elementary operations on the rows of a matrix. Gauss reduction method for the rank of a matrix and for the resolution of linear systems. Homogeneous linear systems. Determinant of a matrix. Inverse matrix, Gauss-Jordan method.

Vectors. Vector spaces. Linear dependence between vectors. Generators of a vector space. Base of a vector space. Scalar product. Orthogonality between vectors. Orthogonal basis. Vector product. Eigenvectors. Eigenspaces.

Analytic geometry in space. Equation of a plane, line equation in parametric and Cartesian form. Orthogonality and parallelism between planes and lines. Point to line distance. Point-plane distance.

Readings/Bibliography

The teaching material needed to prepare for the exam will be provided by both professor (slides and exercise sheets related to the topics covered). This teaching material will be uploaded to the Virtual platform during the course.

For further deepening, a list of auxiliary texts for possible consultation is provided below:

• J. Stewart, Calculus - Early Trascendentals, 8th Edition, Cencage Learning, 2016
• M. Bramanti, C. D. Pagani, S. Salsa: Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli.
• M. Barnabei, F. Bonetti, Sistemi lineari e matrici, Pitagora Editrice, Bologna
• Daniele Ritelli, Massimo Bergamini, Anna Trifone, Fondamenti di Matematica, Zanichelli
• Daniele Ritelli, Lezioni di Analisi Matematica III Edizione. Esculapio 2019. ISBN: 9788874888870

Teaching methods

During lectures, the focus is primarily on the applications of mathematics, with special emphasis on environmental sciences.

Topics are presented along with many examples and exercises.

Assessment methods

The knowledge and skills to be acquired are verified through a written test. The test consists of exercises, as well as one or two open questions of a more theoretical nature. The maximum score for the entire written test is 32. The score for completely incorrect or absent answers is zero, no negative scores are foreseen.

The time available to the student for the written test is 2 hours. During the test, the use of support material such as textbooks, notes, summary sheets, diagrams, cell phones (even with calculator apps), smartphone, tablet, computer media is not permitted. The use of a simple personal calculator is allowed. The use of AI is prohibited. Any use constitutes a violation of academic integrity.

The maximum score that can be obtained by providing all correct and complete answers is therefore 30 cum laude, corresponding to a score of 31 or 32 obtained in the written test. 

All exercises are comparable (by type and difficulty level) with those done during the classroom exercises and with the supplementary exercises made available by the teachers during the course by uploading them to the Virtual platform.

There is the possibility of taking an oral test following the written test. Students are admitted to the oral test with a score obtained in the written test of 15 or higher. This oral test may confer a positive or negative score of up to 3 points and is optional if the score obtained in the written test is greater than or equal to 18; it is mandatory for passing the exam if the score obtained in the written test is 15, 16 or 17. With a score of less than 15 in the written test, you are not admitted to the oral examination, the examination grade is insufficient and is recorded as RE (rejected).

Class attendance is very important in the learning process and strongly recommended, but it does not influence the evaluation process in any way.

There are no papers to be delivered before the exam date.

To participate in the exam, it is necessary to register via AlmaEsami for the chosen exam session, by the scheduled deadlines.

The student who is registered on the AlmaEsami exam list on the day registration closes but does not show up to take the exam or notifies the teacher or unregisters on AlmaEsami, or the student who during the written exam expresses the desire to withdraw, requesting the professor to write RT on the sheets submitted, will be given the grade RT (withdrawn).

The student who submits his/her written test for correction without declaring the desire to withdraw and receives an insufficient grade (grade lower than 18/30) will be given the grade RE (rejected).

Students with specific learning disorders (SLD) and/or  temporary or permanent disabilities: please, contact the responsible University office (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students ) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the professors, who will assess the appropriateness of the adjustments, taking into account the teaching objectives of the course.

Teaching tools

Slides and other material provided in electronic format (exercise sheets, etc.).

Office hours

See the website of Silvia Tozza

See the website of Silvia Foschi