Real numbers. Sequences and function. Limits of sequences and function. Elementary functions. Continuous functions. Differentiable functions and their properties. Local maxima and minima. Graphics of functions. Taylor's formula. Riemann's integral. Integrability of continuous functions. Newton-Leibniz Formula. Antiderivatives. Numerical series. Improper integrals.
The course intends to give the fundamental concepts of differential and integral calculus for real function in one real variable: continuity, differentiability, polynomial approximation, Riemann’s integral, fundamental theorem of integral calculus. The “continue” point of view is joined to the “discrete” with the study of sequence and numeric series concepts.
At the end of the course the student learns how to apply the tools of calculus to the study of functions of one real variable, the research of maxima and minima, function approximation, evaluations of areas and volumes.
Prerequisites
Trigonometry and basic inequalities
Teaching Methods
CFU: 12
Contact hours for: Lectures (hours): 108
Intermediate examinations: 9
Further information
Office hours: to be decided
e-mail:
villari@math.unifi.it
focardi@math.unifi.it
Website:
http://web.math.unifi.it/users/villari
http://web.math.unifi.it/users/focardi/
Type of Assessment
Written test followed by an oral test
Course program
Real numbers: definition and properties. Upper and lower bound of a set.
Numeric sequences: limit definition, limit uniqueness, comparison theorems, undetermined forms, and remarkable limits. Neper’s number.
Functions in one real variable: definitions of domain, codomain, injectivity, surjectivity, invertibility. Odd, even and periodic functions. Continuous functions: definition and main theorems (Weierstrass, theorem of zeros and intermediate values).
Derivatives: definition and main theorems (Fermat, Rolle, and Lagrange).
Antiderivatives and integration methods.
Riemann's integrals. Fundamental theorem of calculus.
Areas of plane regions and volumes of solids. Taylor's formula.
Improper integrals. Numerical series.