Università Roma Tre

Part 1: School Skills Review.
Real numbers and their subsets (N, Z, Q).
Roots and properties of rational powers.
Inequalities (also graphic resolution).
Fundamental properties of exponential, logarithmic, trigonometric and inverse trigonometric functions.

Part 2: Introduction to the concept of limit, continuity and differentiability through definitions, examples and exercises
Definition of limit for functions from R to R.
Calculation of delta as a function of epsilon in simple cases.
Fundamental properties of limits: algebra of limits and computation of finite limits.
Infinite limits, limit of sequences.
Extended limits algebra: extension of the calculus of limits.
Continuous functions and points of discontinuity.
Derivative: definition and rules of derivation (statements). Calculation of derivatives.
Relation between derivative and monotony.
Convexity: definition and criteria for C^1 and C^2 functions.
Applications to the qualitative study of function graphs.

Part 3: Introduction to the concept of integral and series through definitions, examples and exercises.
Definition of Riemann integral and its fundamental properties (linearity, invariance by translation, positivity). Calculation of simple integrals using the definition.
Illustration of the fundamental theorem of integral calculus.
Calculation of Primitives: main methods (substitution, integration by parts); integration of rational functions and other special classes.
Numerical series. Convergence criteria: statements and applications.
Improper integrals. Convergence criteria: statements and applications.

Part 4: Elementary solution methods of ordinary differential equations
Solution methods for special classes of ordinal differential equations (EDO) including:
linear first order, separation of variables, second order with constant coefficients, etc.

"Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R", L. Chierchia, McGraw-Hill Education
"Analisi Matematica 1", E. Giusti, Bollati Boringhieri

Exercise book:
"Esercizi e complementi di Analisi Matematica, Volume Primo", E. Giusti, Bollati Boringhieri
"Esercizi e problemi di Analisi Matematica", B.P. Demidovich, Editori Riuniti

Numbers. Inequalities. Exponential functions and logarithms. Trigonometric functions and their inverses (applies to OFA recovery);
Limits (sequences, functions) and continuity. Derivatives. Graphs of functions;
Integrals. Series and improper integrals;
Differential equations: first order linear equations, with separation of variables, second order equations with constant coefficients, non-autonomous.

"Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R", L. Chierchia, McGraw-Hill Education
"Analisi Matematica 1", E. Giusti, Bollati Boringhieri