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

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.

Sequences and limits of sequences. Definition of sequence and of its limit, monotone sequences. Computation of limits. Some relevant limits. Comparison theorems.

Numerical series. Partial sums and convergin series. Necessary condition for convergence. Non-negative-terms series: convergence criteria. Variable-signed-terms series: absolute convergence and convergence. Geometric series.

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.

Analytic geometry on the plane. Points on the plane: vectors and coordinates. Operations between vectors, computation of distance and angles. Lines on the plane: equations, parallel and crossing lines. Circles on the plane.

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

Readings/Bibliography

Teaching material provided by the teacher.

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

Exercises: M. Boella: Analisi matematica 1 e algebra lineare. Eserciziario. Pearson

Teaching methods

The lessons are designed to highlight the applicative aspects of the subject, particularly in the field of application of greatest interest for the Architecture-Engineering course of study.

Topics are presented along with many examples and exercises.

 

Assessment methods

Written test consisting of exercises, as well as one or two more theoretical questions. The total time available for the written test is two hours.
All exercises are comparable (by type and level of difficulty) with those carried out during classroom exercises and with the supplementary exercises made available by the teacher during the course.

There is the possibility of taking an oral test following the written test. This oral test can award a positive or negative score and is optional if the score obtained in the written test is greater than or equal to 18.

Teaching tools

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

Students who need compensatory tools for reasons related to disabilities or specific learning disorders (SLD) can directly contact the Service for Students with Disabilities ( disabilita@unibo.it [mailto:disabilita@unibo.it] ) and the Service for Students with learning disabilities ( dsa@unibo.it [mailto:dsa@unibo.it] ) to agree on the adoption of the most appropriate measures.

Office hours

See the website of Silvia Tozza