PRINCIPLES OF MATHEMATICS
Learning outcomes of the course unit
To supply the basic tools of infinitesimal calculus. To introduce the student to the concepts of Mathematical Analysis.
Course contents summary
Sets, subsets, power set. Principle of induction and applications. Equivalence and order relations: raising operators, lowering operators, narrowness. Functions and their opposites. Injectivity, surjectivity. Images and counter images. Sequences. Monotone functions. Composition of functions and invertibility. Digression on numbers: rational numbers and real numbers and their properties. Ordered fields and absolute value: first properties. Powers with real exponent. Graphs of f(x), kf(x), f(x+h), f(x)+h, |f(x)|. Graphs of elementary functions: power, root, exponential, logarithmic function. Revision of trigonometry. Functions of a real variable. Limits, continuity, derivatives, differentials. Monotone functions. Fermat, Rolle, Cauchy, Lagrange theorems (with demonstrations). De l'Hospital's rule. Function graphs. Infinitesimals and infinites. Taylor-MacLaurin polynomials with Peano remainder. Definite and indefinite integrals. Integration methods. Fundamental theorem of integral calculus. Integral function. Geometric applications. Improper integrals. Convergence criteria. Length of arches of plane curves.
Stoka, M: Corso di Matematica, CEDAM. Stoka-Pipitone, Esercizi e problemi di matematica, CEDAM
Theoretical lectures and exercises