90925 - ELEMENTS OF MATHEMATICS

Conoscenze e abilità da conseguire

Al termine del corso lo studente conosce: - gli elementi fondamentali della cultura matematica; - le procedure matematiche necessarie per affrontare i corsi successivi. Più precisamente, lo studente conosce: 1. le procedure algebriche per risolvere disequazioni; 2. le funzioni elementari e semplici trasformazioni geometriche; 3. le più importanti procedure del calcolo differenziale; 4. elementi fondamentali di statistica e probabilità.

Contenuti

Main Contents

Classification of numbers: the sets N, Q, and R, their essential features, and their cardinality. Numerically equivalent sets. Countably and uncountably infinite sets.

Permutations, dispositions (with and without repetition), and combinations.

Equations and inequalities of different kinds: irrational, with absolute value, exponential, and logarithmic.

Continuous mathematics: domains, ranges, and graphs of elementary functions.

Bounded and unbounded functions, even and odd functions, injectivity.

Geometric transformations on functions: symmetries and translations.

Mathematical analysis: some remarkable limits and indeterminate forms. Asymptotes.

Continuity and derivability: derivatives of elementary functions and applications.

Plan of function investigation, convexity and inflexion points included.

Basic algorithms for the computation of primitive functions.

Definite integrals: Newton-Leibniz formula for continuous functions.

Further Contents (to get full marks)

Basic probability: binomial probability formula for Bernoulli trials.

Inverse functions.

Riemann’s improper integrals.

The sum of infinite series: geometric series.

Solving inequalities graphically.

To get full marks “with honour”

Prove that 0! = 1.

Prove that Q is denumerable.

Prove that √2 is not rational.

Prove that the set of primes does not have maximum.

Prove that R is denumerable.

Prove that .

Prove that the derivative of y = √x by solving an indeterminate form.

Prove Torricelli-Barrow-Newton’s theorem.

Testi/Bibliografia

Warner & Costenoble, FINITE MATHEMATICS AND APPLIED CALCULUS, Brooks/Cole, 2017 (OR other editions - some copies available in the university library)

Metodi didattici

Recurring sequences of short lectures and exercises

Modalità di verifica e valutazione dell'apprendimento

Exam

The exam lasts 2 hours, and consists of a hierarchical series of exercises.

The first part deals with the most elementary items of the main contents: it allows students to get a passing grade. The second part allows students to get higher marks.

Strumenti a supporto della didattica

Brief handouts dealing with theoretical memos and exercises

Orario di ricevimento

Consulta il sito web di Stefano Bordoni