Università Roma Tre

Number sets (N, Z, Q and R), axiomatic construction of R via supremum, Archimedean property, density of Q in R, construction of N in R and the inductive method, binomial formula and combinatorial calculus, real powers, the Bernoulli inequality; topological concepts in R (accumulation and isolated points, open/closed sets and characterization, closure of a set) and Bolzano-Weierstrass theorem; complex numbers, polar representation and n-roots of unity; real functions with a real variable, domain, image and inverse functions; limits for functions and properties, limits of monotone functions; limits for sequences, special limits, the Napier number, the bridge theorem,
limsup/liminf, sequences and topology, compact sets and characterization; continuous functions and their properties, continuity of elementary functions, types of discontinuities and monotone functions, fundamental theorems on continuous functions (zeroes, intermediate values, Weierstrass); derivative of a function and properties, derivatives of elementary functions, the fundamental theorems of differential calculus (Fermat, Rolle, Cauchy, Lagrange, de l'Hopital, Taylor's formula), monotonicity and sign of the derivative, degenerate local maxima/minima, convex/concave functions; graph of a function; Riemann integration and properties, integrability of continuous functions, primitives for elementary functions, I and II fundamental theorems of integral calculus, change of variables and integration by parts, rational functions, some special change of variables; numerical series and convergence, geometric series, convergence criteria for positive series (comparison, asymptotic comparison, n-th root, ratio, condensation) and for general series (absolute convergence, Leibniz); Taylor series, series of some elementary functions; improper integrals.

"Calcolo", P. Marcellini, C. Sbordone, editore Liguori
"Esercitazioni di Matematica: vol. 1.1 e 1.2", P. Marcellini, C. Sbordone, editore Liguori

The numbers refer to the chapters and paragraphs of the textbook:
Calcolo of P. Marcellini and C. Sbordone.

1) Real numbers and functions

Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).

2) Complements to real numbers

Maximum, minimum, supremum, infimum.

7) Succession limits

Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).

8) Function limits. Continuous functions

Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).

9) Additions to the limits

The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).

10) Derivatives

Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).

11) Applications of derivatives. Function study

Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).

14) Integration according to Riemann

Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).

15) Undefined integrals

The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).

16) Taylor's formula

Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).

17) Series

Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).

P. Marcellini, C. Sbordone, Calcolo, Ed. Liguori, 1992
S. Lang, A First Course in Calculus, Springer Ed.
L.Chierchia, Corso di Analisi - Prima parte, McGraw Hill (2019)

Exercises.

The numbers refer to the chapters and paragraphs of the textbook:
Calcolo of P. Marcellini and C. Sbordone.

1) Real numbers and functions

Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).

2) Complements to real numbers

Maximum, minimum, supremum, infimum.

7) Succession limits

Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).

8) Function limits. Continuous functions

Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).

9) Additions to the limits

The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).

10) Derivatives

Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).

11) Applications of derivatives. Function study

Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).

14) Integration according to Riemann

Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).

15) Undefined integrals

The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).

16) Taylor's formula

Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).

17) Series

Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).

P. Marcellini, C. Sbordone, Calcolo, Ed. Liguori, 1992
S. Lang, A First Course in Calculus, Springer Ed.
L.Chierchia, Corso di Analisi - Prima parte, McGraw Hill (2019)