| MATHEMATICAL ANALYSIS I
(objectives)
Allow the acquisition of the deductive logic method and provide the basic mathematical tools of the differential and integral calculus. Each topic will be rigorously introduced and treated, performing, sometimes, full proofs and also making a strong connection with the physical meaning, the geometric interpretation and the numerical application. A proper methodology and a reasonable skill in the use of concepts of the integro-differential calculus and related results will enable the students to possibly face in an easily way the more applied topics that will be developed in the later courses.
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Code
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20810231 |
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Language
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ITA |
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Type of certificate
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Profit certificate
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Credits
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12
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Scientific Disciplinary Sector Code
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MAT/05
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Contact Hours
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108
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Type of Activity
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Basic compulsory activities
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Group: CANALE 1
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Teacher
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FEOLA ROBERTO
(syllabus)
Real numbers and functions, set theory, induction principle, infimum and supremum.
Sequences, definition of limit, operations with limits, comparison theorems, infinitives of increasing order.
Limits of function, continuity, link with the limits of sequences, theorems on continuous functions.
Derivatives, geometric meaning, theorems on differentiable functions, relative maximums and minimums, applications to the study of functions. Indefinite integrals, integration by parts and by substitution, definite integrals, fundamental theorem of integral calculus, improper integrals. Numerical series, simple and absolute convergence, convergence criteria.
Series of functions, point and total convergence. Complex numbers.
(reference books)
Marcellini, Sbordone - Elements of Mathematical Analysis 1,2.
Marcellini, Sbordone - Exercises in mathematics Vol. 1, 2.
Bertsch, Dall'Aglio, Giacomelli - Epsilon 1, A first course in Mathematical Analysis . McGraw Hill
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Dates of beginning and end of teaching activities
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From 26/09/2022 to 13/01/2023 |
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Delivery mode
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Traditional
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Attendance
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not mandatory
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Evaluation methods
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Written test
Oral exam
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Teacher
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ROMOLI MATTEO
(syllabus)
Real numbers and functions, set theory, induction principle, infimum and supremum.
Sequences, definition of limit, operations with limits, comparison theorems, infinitives of increasing order.
Limits of function, continuity, link with the limits of sequences, theorems on continuous functions.
Derivatives, geometric meaning, theorems on differentiable functions, relative maximums and minimums, applications to the study of functions. Indefinite integrals, integration by parts and by substitution, definite integrals, fundamental theorem of integral calculus, improper integrals. Numerical series, simple and absolute convergence, convergence criteria.
Series of functions, point and total convergence. Complex numbers.
(reference books)
Marcellini, Sbordone - Elementi di Analisi matematica 1,2.
Marcellini, Sbordone - Esercitazioni di matematica Vol. 1, 2.
Bertsch, Dall'Aglio, Giacomelli - Epsilon 1, Primo corso di Analisi Matematica . McGraw Hill
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Dates of beginning and end of teaching activities
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From 26/09/2022 to 13/01/2023 |
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Delivery mode
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Traditional
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Attendance
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not mandatory
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Evaluation methods
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Written test
Oral exam
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Group: CANALE 2
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Teacher
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SCOPPOLA ELISABETTA
(syllabus)
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).
(reference books)
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)
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Dates of beginning and end of teaching activities
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From 26/09/2022 to 13/01/2023 |
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Delivery mode
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Traditional
At a distance
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Attendance
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not mandatory
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Evaluation methods
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Written test
Oral exam
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Teacher
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REUVERS Robin Johannes Petrus
(syllabus)
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).
(reference books)
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)
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Dates of beginning and end of teaching activities
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From 26/09/2022 to 13/01/2023 |
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Delivery mode
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Traditional
At a distance
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Attendance
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not mandatory
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Evaluation methods
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Written test
Oral exam
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Università Roma Tre