CALCULUS I

Ingegneria Gestionale CALCULUS I

0612600001
DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
EQF6
INDUSTRIAL ENGINEERING AND MANAGEMENT
2022/2023

OBBLIGATORIO
YEAR OF COURSE 1
YEAR OF DIDACTIC SYSTEM 2018
AUTUMN SEMESTER
CFUHOURSACTIVITY
990LESSONS


Objectives
THE COURSE OF MATHEMATICS I AIMS AT THE KNOWLEDGE OF BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND OF CALCULUS, AND TO THEIR PHYSICAL AND GEOMETRICAL INTERPRETATIONS. IT HAS THE FOLLOWING EDUCATIONAL OBJECTIVES:

-) KNOWLEDGE AND UNDERSTANDING

THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS, IN PARTICULAR SETS OF NUMBERS, REAL-VALUED FUNCTIONS, SEQUENCES OF REAL NUMBERS, LIMITS OF REAL-VALUED FUNCTIONS, CONTINUOUS FUNCTIONS, DERIVATIVES OF REAL-VALUED FUNCTIONS, THE FUNDAMENTAL THEORY OF DIFFERENTIAL CALCULATION, THE GRAPHIC STUDY OF A FUNCTION, THE INTEGRALS OF THE FUNCTIONS OF A VARIABLE AND NUMERICAL SERIES OF REAL NUMBERS, THE CANONICAL FORM OF CONICAL SECTIONS.


-) APPLYING KNOWLEDGE AND UNDERSTANDING

THE STUDENT HAS TO DEVELOP IN A RIGOROUS AND COHERENT WAY A MATHEMATICAL ARGUMENT, APPLY THE THEOREMES AND THE STUDIED RULES TO SOLVING PROBLEMS. HE HAS TO BE ABLE TO MAKE CALCULATIONS WITH LIMITS, DERIVATIVES AND INTEGRALS (BOTH UNDEFINED AND DEFINED).

-) LEARNING SKILLS

THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.
Prerequisites
FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS OF THE COURSE, STUDENTS ARE REQUIRED TO HAVE THE FOLLOWING PREREQUISITES:
-KNOWLEDGE OF ALGEBRA, WITH PARTICULAR REFERENCE TO: ALGEBRAIC, LOGARITHMIC AND EXPONENTIAL EQUATIONS AND INEQUALITIES.
-KNOWLEDGE RELATED TO TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC TRIGONOMETRIC FUNCTIONS.
Contents
NUMERICAL SETS (2 HOURS): INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS.

COMPLEX NUMBERS (7 HOURS). OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’S FORMULA. N-TH ROOTS.

REAL FUNCTIONS (10 HOURS): DOMAIN AND CODOMAIN. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. MAIN ELEMENTARY FUNCTIONS.

NUMERICAL SEQUENCES (6 HOURS): BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.

LIMITS OF A FUNCTION (10 HOURS): UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.

CONTINUOUS FUNCTIONS (10 HOURS): CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM, ZEROS THEOREMS).

DERIVATIVE OF A FUNCTION (8 HOURS): DERIVABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER-ORDER DERIVATIVES. DERIVATIVE OF A FUNCTION AND ITS GEOMETRIC MEANING.
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, AND LAGRANGE THEOREMS, THEIR COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.

THE STUDY OF A FUNCTION (8 HOURS): ASYMPTOTES, MAXIMA AND MINIMA. CONCAVE AND CONVEX BEHAVIOR, INFLECTION POINTS. THE GRAPH.

INTEGRATION OF REAL FUNCTIONS (15 HOURS): PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.

NUMERICAL SERIES (6 HOURS): INTRODUCTION TO MAIN NUMERICAL SERIES.

CONIC SECTIONS (8 HOURS): CANONICAL FORMS OF CIRCLES, ELLIPSES, HYPERBOLAS, AND PARABOLAS
Teaching Methods
NUMERICAL SETS (2 HOURS): INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS.

COMPLEX NUMBERS (7 HOURS). OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’S FORMULA. N-TH ROOTS.

REAL FUNCTIONS (10 HOURS): DOMAIN AND CODOMAIN. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. MAIN ELEMENTARY FUNCTIONS.

NUMERICAL SEQUENCES (6 HOURS): BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.

LIMITS OF A FUNCTION (10 HOURS): UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.

CONTINUOUS FUNCTIONS (10 HOURS): CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM, ZEROS THEOREMS).

DERIVATIVE OF A FUNCTION (8 HOURS): DERIVABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER-ORDER DERIVATIVES. DERIVATIVE OF A FUNCTION AND ITS GEOMETRIC MEANING.
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, AND LAGRANGE THEOREMS, THEIR COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.

THE STUDY OF A FUNCTION (8 HOURS): ASYMPTOTES, MAXIMA AND MINIMA. CONCAVE AND CONVEX BEHAVIOR, INFLECTION POINTS. THE GRAPH.

INTEGRATION OF REAL FUNCTIONS (15 HOURS): PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.

NUMERICAL SERIES (6 HOURS): INTRODUCTION TO MAIN NUMERICAL SERIES.

CONIC SECTIONS (8 HOURS): CANONICAL FORMS OF CIRCLES, ELLIPSES, HYPERBOLAS, AND PARABOLAS
Verification of learning
THE EXAM IS COMPOSED BY A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT HAS TO PASS BOTH THE WRITTEN AND ORAL TEST.

THERE WILL BE IN-COURSE WRITTEN TESTS, ON THE LATEST TOPICS SEEN DURING THE LECTURES. THE STUDENTS WHO PASS THOSE TESTS WILL BE EXONERATED FROM THE WRITTEN TEST.

WRITTEN TEST: IT CONSISTS IN SOLVING PROBLEMS SIMILAR TO THE ONES WHICH HAVE BEEN STUDIED DURING THE COURSE. IN MORE DETAILS, THERE WILL BE THE FOLLOWING TYPES OF EXERCISES: DETERMINATION OF THE DOMAIN OF A SCALAR FUNCTION (AT LEAST 6 POINTS); SOLUTION OF AN EQUATION IN THE COMPLEX DOMAIN (AT LEAST 3 POINTS); SOLUTION OF A LIMIT DISPLAYING AN INDETERMINATE FORM (AT LEAST 6 POINTS); STUDY OF A SCALAR FUNCTION (AT LEAST 6 POINT); COMPUTATION OF (DEFINITE/INDEFINITE) INTEGRALS (AT LEAST 6 POINTS); CHECK OF A NUMERICAL-SERIES' BEHAVIOR (AT LEAST 3 POINTS). SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30 AND IS THE SUM OF THE SCORES OBTAINED IN EACH EXERCISES ABOVE. THOSE SCORES DEPEND ON THE STUDENT'S ABILITY IN APPLYING OWN KNOWLEDGE. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED.

ORAL INTERVIEW: IT IS SUBJECTED TO BEING SELECTED IN THE PREVIOUS WRITTEN TEST. IT NORMALLY TAKES 20 MINUTES. IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING.

FINAL EVALUATION: THE FINAL MARK, EXPRESSED ON A SCALE FROM 18 TO 30 (POSSIBLY WITH LAUDEM), DEPENDS ON THE MARK OF THE WRITTEN EXAM, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.
Texts
-) ONLINE NOTES.

-) THEORY
- P. MARCELLINI - C. SBORDONE, “ANALISI MATEMATICA UNO “, LIGUORI EDITORE (1996)

-) EXERCISES
- P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA I - VOL. 1 E 2“, LIGUORI EDITORE (2016)
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