CFU
9
Length
14 Weeks
Semester DD
First
Foundations: the limits of "school level" set theory; preliminaries from logic; modern set theory; relations; natural, integer, rational and real numbers.
Functions and sequences: real elementary functions of a real variable; notion of sequence; notable sequences, factorial, binomial coefficients.
Limits: notion of limit; properties of limits; notable limits; some notions of elementary topology.
Numerical series: definition and elementary properties; series with positive terms; root and ration criteria; absolute convergence; series with alternating signs.
Continuous functions: definitions and elementary properties; theorem of the intermediate values and its consequences; continuity of the inverse function; continuous functions in a closed bounded interval.
Derivatives: definition and geometric interpretation; elementary properties of derivatives; differentiable functions in an interval; function approximation, Taylor polynomials and Taylor formula; monotony and extremes; convexity; general method for the study of a function; hyperbolic functions; exponential, sine and cosine in the complex plane.
Integrals: definition of Riemann integral; integrable functions; properties of the Riemann integral; integration methods; improper integrals; application of integral calculus; basic notions on ordinary differential equations.
LEARNING OUTCOMES: to acquire knowledge of the basic concepts of limits, derivatives, and integrals of functions of a real variable, and differential equations; to be able to apply these concepts to solve s: knowledge of the basic concepts of mathematical analysis, starting from the importance of the concept of continuum in physics up to the mastery of the basic tools for integral and differential calculus in one dimension.
KNOWLEDGE AND UNDERSTANDING: understand formal reasoning, distinguishing between the heuristic and rigorous vision of a concept.
ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: knowing how to apply the notions learned to concrete applications and problems posed by physics.
MAKING JUDGMENTS: being able to evaluate the mathematical techniques learned necessary to solve the concrete problems posed by physics.
COMMUNICATION SKILLS: explaining and arguing the solution to problems; also be able to discuss and correctly reproduce demonstrations of basic results relating to mathematical analysis.
LEARNING SKILLS: knowing how to identify solution strategies in situations similar to those addressed in the course.