# Ingegneria Gestionale | CALCULUS I

## Ingegneria Gestionale CALCULUS I

 0612600001 DIPARTIMENTO DI INGEGNERIA INDUSTRIALE EQF6 INDUSTRIAL ENGINEERING AND MANAGEMENT 2019/2020

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 9 90 LESSONS BASIC COMPULSORY SUBJECTS

 ANTONIO SELLITTO T ISABELLA CARLOMAGNO
Objectives
THE COURSE AIMS AT 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.

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

REAL FUNCTIONS (16 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, ZEROS, AND BOLZANO THEOREMS. UNIFORM CONTINUITY.

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 (16 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.
Teaching Methods
COMPULSORY ATTENDANCE. LECTURES ARE IN ITALIAN.

THE COURSE CONSISTS OF THEORETICAL LECTURES, DEVOTED TO THE EXPLANATION OF ALL THE COURSE CONTENTS AND CLASSROOM PRACTICE, PROVIDING THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES.
Verification of learning
THE EXAM IS COMPOSED BY A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT IS REQUIRED TO PASS BOTH THE WRITTEN AND ORAL TESTS.

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 STUDIED DURING THE COURSE . SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED

ORAL INTERVIEW: 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
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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