00674 - Mathematics

Learning outcomes

At the end of the course the student becomes acquainted with the basic concepts and tools of mathematical analysis of real functions of a real variable; the student  will also become acquainted with the basic concepts of matrix theory and linear algebra. In particular, the student will be able to: - compute limits, derivatives and Taylor polynomials of a function - study and plot functions - compute integrals and generalized integrals of a function - solve systems of linear equations by using vectors and matrices - recognize linearly independent sets and subspaces - compute scalar products, norms, orthogonal projections - diagonalize square matrices.

Course contents

The course will be delivered jointly by Francesco Regonati and Luca Vincenzo Ballestra. The contents of the course are listed below, with the indication of the corresponding lecturer.

1. Basic notions about sets. Some elementary logics. The summation symbol and binomial coefficients. Ordered fields. Real numbers; least upper bound and the continuity axiom; absolute value; intervals. Radicals, powers, logarithms. Infinite sets. The induction principle. (Regonati)
2. The notion of a function. Real functions of a real variable; bounded, symmetric, monotone, periodic functions. Elementary functions; power, exponential, logarithmic, trigonometric functions; operations on graphs. Composition of functions; inverse function; inverse trigonometric functions. (Regonati)
3. Sequences; definition of the limit; monotone sequences; limit computations; the number "e". Numerical series; definition and first examples; non negative series. Limits of functions. Limit computations; fundamental properties of limits; some special limits; comparison and asymptotic analysis. (Regonati) Continuity, asymptotes. Continuous functions on a compact set and on an interval. (Ballestra)
4. Derivative and tangent line. Derivatives of elementary functions. Angular points, vertical tangent lines. Algebra of derivatives. Derivative of a composite function. Maxima and minima. Stationary points. Mean value theorem. Monotonicity test. De l'Hopital theorem. Second-order derivative, concavity and convexity. Studying and plotting functions. Differential and linear approximation. Taylor-MacLaurin polynomials with Peano's remainder. (Ballestra)
5. The integral as a limit of sums; classes of integrable functions. Properties of the integral. The fundamental theorem of integral calculus. Computation of indefinite and definite integrals; basic integrals; integration of rational functions; integration by parts; integration of discontinuous functions. Generalized integrals; integration of unbounded functions and of functions defined on unbounded intervals; integrability criteria. Integral function. (Regonati)
6. Plane and space vectors; fundamental operations on vectors; scalar product. Straight lines and planes in space. The vector space R^n; abstract vector spaces; vector subspaces, linear independence, bases and dimension. Scalar product in R^n; vector spaces with scalar product; norm, orthogonality, orthonormal bases, orthogonal projections on subspaces. The notion of linearity. Matrices and linear transformations; matrix algebra; matrix rapresentation of linear transformations; determinants; matrix rank; inverse matrix. Systems of linear equations; Cramer rule; image and kernel of a linear trasformations from R^m to R^n; Rouchè-Capelli theorem; Gaussian elimination. Diagonalizable matrices; eigenvalues and eigenvectors of a matrix; conditions of diagonalizability. (Regonati)

Readings/Bibliography

M. Bramanti, C. D. Pagani, S. Salsa: Analisi matematica 1 con elementi di geometria e algebra lineare. Ed. Zanichelli, Bologna, 2014

Teaching methods

Since this is a traditional course in mathematics, lessons will be taught at the blackboard.

Assessment methods

The learning outcomes are verified through both a written and an oral exam.

The written exam takes two hours and consists of some exercises where students must demonstrate their practical skills. The use of books or drafts is not permitted. Non-programmable pocket calculators are allowed.

The oral exam aims at verifying the understanding of the theory.

The final grade will be established on the basis of the grade of the written exam and of the judgement of the oral exam.

Teaching tools

The lectures will consist of theory, examples and exercises. During the course some teaching materials will be provided by the lecturers as integration/completion of the textbook.

Finally, the course will be preceded by a crash-course, such to provide students with the necessary basic mathematical knowledge. All the students are strongly encouraged to attend it.

Office hours

See the website of Francesco Regonati

See the website of Luca Vincenzo Ballestra