# Fisica | CALCULUS I

## Fisica CALCULUS I

 0512600001 DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" EQF6 PHYSICS 2020/2021

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 10 80 LESSONS BASIC COMPULSORY SUBJECTS MAT/05 2 24 EXERCISES BASIC COMPULSORY SUBJECTS
 PAOLA CAVALIERE T
Objectives
KNOWLEDGE AND UNDERSTANDING:

THE MAIN OBJECTIVE OF THE COURSE IS FOR STUDENTS TO LEARN THE BASICS OF THE CALCULUS OF FUNCTIONS OF ONE VARIABLE. STARTING FROM BASIC SET THEORY, THE COURSE WILL GIVE STUDENTS A STRONG FOUNDATION OF BASIC MATH SKILLS, CONCEPTS, VOCABULARY, AND PROBLEM-SOLVING STRATEGIES FOR THE STUDY OF REAL ONE-VARIABLE FUNCTIONS AS WELL AS SERIES OF REAL NUMBERS.
THE COURSE IS DESIGNED TO ENABLE STUDENTS TO
-INTERPRET THE CONCEPTS OF CALCULUS ALGEBRAICALLY, GRAPHICALLY AND VERBALLY;
-DEVELOP THEIR ABILITY TO THINK IN A CRITICAL MANNER,
-TO FORMULATE AND DEVELOP MATHEMATICAL ARGUMENTS IN A LOGICAL MANNER;
-IMPROVE THEIR SKILLS IN ACQUIRING NEW UNDERSTANDING AND EXPERIENCE;
-ANALYZE MATHEMATICAL PROBLEMS IN CALCULUS AND SOLVE IT USING A WIDE ARRAY OF TOOLS.

APPLYING KNOWLEDGE AND UNDERSTANDING:

HAVING COMPLETED THE ESSENTIAL READING AND ACTIVITIES, STUDENTS SHOULD:
- HAVE A GOOD KNOWLEDGE OF BASIC MATHEMATICAL CONCEPTS AND SKILLS IN CALCULUS;
- BE ABLE TO DEMONSTRATE THE ABILITY TO SOLVE UNSEEN MATHEMATICAL IN CALCULUS AND RELATED APPLIED FIELDS;
- UNDERSTAND AND COMMUNICATE CLEARLY AND EFFECTIVELY THE PRINCIPLES OF CALCULUS, USING PROPER VOCABULARY AND NOMENCLATURE;
- DEVELOP EFFECTIVE STUDY SKILLS IN ORDER TO MASTER COURSE CONTENT AND OBJECTIVES;
- THINK CRITICALLY IN READING AND WRITING MATHEMATICS, DETERMINING IN PARTICULAR IF CONCLUSIONS OR SOLUTIONS ARE REASONABLE.
Prerequisites
THE PREREQUISITES ARE HIGH SCHOOL ALGEBRA, GEOMETRY AND TRIGONOMETRY.
Contents
FOUNDATIONS (T:5 HOURS) - LOGIC, SET NOTATIONS, CARTESIAN PRODUCT OF SETS, RELATIONS. FUNCTIONS. EQUIVALENCE AND ORDER RELATIONS.

THE REAL NUMBER SYSTEM (T: 7 HOURS)- THE AXIOMS OF THE REAL NUMBER SYSTEM. THE SUBSETS N OF NATURAL NUMBERS, THE PRINCIPLE OF MATHEMATICAL INDUCTION AND ITS APPLICATIONS. THE SETS OF THE INTEGERS Z AND OF THE RATIONAL NUMBERS Q. GEOMETRIC REPRESENTATION OF R AND OF RXR.

REAL FUNCTIONS (T: 6 HOURS)– BOUNDS AND GRAPHIC OF ONE-VARIABLE REAL FUNCTION. MONOTONE FUNCTIONS. EVEN, ODD AND PERIODIC FUNCTIONS. REAL SEQUENCES. MONOTONE SEQUENCES. THE NEPER NUMBER E.

ELEMENTARY FUNCTIONS (T: 6 HOURS; E: 6 HOURS) – LINEAR FUNCTIONS. ABSOLUTE VALUE FUNCTION. N-POWER AND N-SQUARE FUNCTIONS. EXPONENTIAL AND LOGARITHMIC FUNCTIONS. REAL POWER FUNCTIONS. TRIGONOMETRIC FUNCTIONS AND THEIR LOCALLY INVERSES.

THE COMPLEX FIELD (T: 4 HOURS; E: 2 HOURS) - DEFINITION, PROPERTIES AND GEOMETRICAL REPRESENTATION. DE MOIVRE FORMULAS. ROOTS OF A COMPLEX NUMBER.

LIMITS OF FUNCTIONS (T: 10 HOURS; E: 6 HOURS) – EXTENDED REAL NUMBERS AND ITS TOPOLOGY. CLOSURE POINTS FOR A SET. CONCEPT OF LIMIT AND ITS PROPERTIES: UNIQUENESS, ALGEBRA OF LIMITS, INDETERMINATE FORMS, COMPARISON THEOREMS. MONOTONE FUNCTIONS AND THEIR LIMITS. COMPOSITE FUNCTIONS AND THEIR LIMITS. LIMITS OF SEQUENCES.

CONTINUOUS FUNCTIONS (T: 4 HOURS; E: 2 HOURS) - CONCEPT OF CONTINUITY AT A POINT. DISCONTINUITY POINTS. THE INTERMEDIATE-VALUE THEOREM AND THE WEIERSTRASS THEOREM. UNIFORMLY CONTINUOUS FUNCTIONS.

THE NOTION OF DERIVATIVE (T: 4 HOURS; E: 2 HOURS) – DEFINITIONS, EXAMPLES AND GEOMETRICAL MEANING. DERIVATION AND ALGEBRAIC OPERATIONS. DERIVATION OF COMPOSITE AND INVERSE FUNCTIONS. DERIVATION OF ELEMENTARY FUNCTIONS.

APPLICATION OF DIFFERENTIAL CALCULUS (T: 6 HOURS; E: 4 HOURS) - QUALITATIVE STUDY OF FUNCTIONS – LOCAL MAXIMUM AND MINIMUM POINTS. THE FERMAT THEOREM. THE THEOREMS OF ROLLE, LAGRANGE AND CAUCHY: RELATIONSHIPS AND CONSEQUENCES. CONVEX AND CONCAVE FUNCTIONS: DEFINITIONS AND PROPERTIES. CRITERIA FOR CONVEXITY. DE L’HOPITAL THEOREMS. QUALITATIVE STUDY OF THE GRAPH OF A FUNCTION. TAYLOR POLYNOMIALS. HYPERBOLIC FUNCTIONS.

RIEMANN INTEGRAL (T: 6 HOURS)– DEFINITIONS AND EXAMPLES. GEOMETRICAL MEANING OF DEFINITE INTEGRAL. MAIN PROPERTIES OF FINITE INTEGRATION. INTEGRABILITY OF CONTINUOUS FUNCTIONS AS WELL AS OF MONOTONE FUNCTIONS. MEAN VALUE THEOREMS.

INDEFINITE INTEGRAL (T: 2 HOURS; E: 6 HOURS) – PRIMITIVE FUNCTIONS. THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. INDEFINITE INTEGRAL AND ITS PROPERTIES. INTEGRATION METHODS: DECOMPOSITION IN SUMS, INTEGRATION BY PARTS AND SUBSTITUTION.

NUMERICAL SERIES (T: 4 HOURS; E: 4 HOURS) – DEFINITIONS, EXAMPLES AND PRELIMINARY RESULTS. SERIES OF POSITIVE AND NEGATIVE TERMS. GEOMETRIC AND HARMONIC SERIES. TESTS FOR CONVERGENCE AND DIVERGENCE. ABSOLUTELY CONVERGENT SERIES. ALTERNATING SERIES AND LEIBNIZ THEOREM.
Teaching Methods
THE COURSE IS STRUCTURED AS A COMBINATION OF LECTURES AND PRATICAL SESSIONS. THE FEEDBACK FROM STUDENTS WILL ALWAYS BE HIGHLY APPRECIATED.
Verification of learning
THE EXAM CONSISTS IN TWO PARTS, A PRELIMINARY WRITTEN TEST AND AN ORAL ONE. THE WRITTEN TEST CONSISTS OF 4/5 EXERCISES RELATED TO THE ARGUMENT OF THE PROGRAM. DURING THE THREE HOUR-LONG WRITTEN EXAMINATION, STUDENTS ARE NOT ALLOWED TO USE TEXTBOOKS, NOTES, SCIENTIFIC CALCULATORS OR OTHER ELECTRONIC DEVICES.
STUDENTS WILL BE EVALUATED ON THEIR ABILITY TO DEVISE, ORGANIZE AND PRESENT COMPLETE SOLUTIONS TO PROBLEMS.
ONLY STUDENTS REACHING A POSITIVE GRADE AT WRITTEN TEST (AT LEAST 16 ON 30) ARE ALLOWED TO TAKE THE ORAL EXAMINATION.
IT MAINLY CONCERNS THE THEORETICAL ASPECTS OF SUBJECT. THE STUDENT HAS TO SHOW TO KNOW THE MAIN TOPICS AND THEIR CONNECTIONS. THE ORAL EXAMINATION IS AIMED TO EVALUATE THE COMPREHENSIVE KNOWLEDGE OF THE MATTER.
THE FINAL GRADE WILL BE DETERMINED AS FOLLOWS: 50% WRITTEN TEST AND 50% ORAL ONE.

THE MINIMUM GRADE OF 18/30 IS OBTAINED WHEN THE STUDENT EXHIBITS A SUFFICIENT KNOWLEDGE OF ALL THE PROGRAM, WITH THE EXCEPTION OF PROOFS.

THE MAXIMUM GRADE OF 30/30 WHEN THE STUDENT EXHIBITS A DEEP AND EXTENDED KNOWLEDGE OF THE THEORY AND THE APPLICATIONS OF ALL THE ARGUMENTS OF THE COURSE.

THE MAXIMUM GRADE CUM LAUDE REQUIRES A KNOWLEDGE OF ALL THE ARGUMENTS WITHIN CONTEXTS GOING BEYOND THE ONES INVESTIGATED IN THE COURSE.

Texts
W. RUDIN: PRINCIPLES OF MATHEMATICAL ANALYSIS, MCGRAW-HILL, 1976, 342 PAGINE, ISBN 00-708-5613-3
T. APOSTOL: MATHEMATICAL ANALYSIS, ADDISON-WESLEY PUBLISHING COMPANY, 1974, 492 PAGES, ISBN 02-010-0288-4

THE STUDENT CAN USE ANY GOOD TEXT OF MATHEMATICAL ANALYSIS CONTAINING THE ARGUMENTS OF THE PROGRAM, SINCE IT IS A STANDARD PROGRAM. STUDENTS ARE URGED TO CHECK IN ADVANCE WITH THE TEACHER THE APPROPRIATENESS OF THE CHOSEN TEXT.