# Fisica | CALCULUS I

## Fisica CALCULUS I

 0512600001 DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" EQF6 PHYSICS 2018/2019

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 12 96 LESSONS BASIC COMPULSORY SUBJECTS
 PAOLA CAVALIERE T
Objectives
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.

KNOWLEDGE AND UNDERSTANDING:

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
UNIT HOURS 8: FOUNDATIONS - LOGIC, SET NOTATIONS. RELATIONS. FUNCTIONS. EQUIVALENCE AND ORDER RELATIONS.

UNIT HOURS 10:THE REAL NUMBER SYSTEM - THE AXIOMS OF THE REAL NUMBER FIELD. THE SUBSETS N OF NATURAL NUMBERS (AND THE PRINCIPLE OF INDUCTION), OF THE INTEGERS Z AND OF THE RATIONAL NUMBERS Q. INTERVALS OF R. GEOMETRIC REPRESENTATION OF R AND OF R2.

UNIT HOURS 4:REAL FUNCTIONS – BOUNDS AND GRAPHIC OF ONE-VARIABLE REAL FUNCTION. MONOTONE FUNCTIONS. EVEN, ODD AND PERIODIC FUNCTIONS. REAL SEQUENCES. MONOTONE SEQUENCES. THE NEPER NUMBER E.

UNIT HOURS 9:ELEMENTARY FUNCTIONS – LINEAR FUNCTIONS. ABSOLUTE VALUE FUNCTION. N-POWER AND N-SQUARE FUNCTIONS. EXPONENTIAL AND LOGARITHMIC FUNCTIONS. REAL POWER FUNCTIONS. TRIGONOMETRIC FUNCTIONS AND THEIR LOCALLY INVERSES.

UNIT HOURS 4: THE COMPLEX FIELD - DEFINITION, PROPERTIES AND GEOMETRICAL REPRESENTATION. DE MOIVRE FORMULAS. ROOTS OF A COMPLEX NUMBER.

UNIT HOURS 12: LIMITS OF FUNCTIONS – EXTENDED REAL NUMBERS AND ITS TOPOLOGY. 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.

UNIT HOURS 6: CONTINUOUS FUNCTIONS. CONCEPT OF CONTINUITY AT A POINT. DISCONTINUITY POINTS. THE WEIERSTRASS THEOREM AND THE INTERMEDIATE-VALUE THEOREM. UNIFORMLY CONTINUOUS FUNCTIONS.

UNIT HOURS 4:THE NOTION OF DERIVATIVE – DEFINITIONS, EXAMPLES AND GEOMETRICAL MEANING. DERIVATION AND ALGEBRAIC OPERATIONS. DERIVATIONS AND COMPOSITE FUNCTIONS. DERIVATION OF ELEMENTARY FUNCTIONS.

UNIT HOURS 8:APPLICATION OF DIFFERENTIAL CALCULUS. QUALITATIVE STUDY OF FUNCTIONS – LOCALLY MAXIMUM AND MINIMUM POINTS. THE FERMAT THEOREM. THE THEOREMS OF ROLLE, LAGRANGE AND CAUCHY: RELATIONSHIPS AND CONSEQUENCES. CONVEX AND CONCAVE FUNCTIONS: DEFINITIONS AND PROPERTIES. DE L’HOPITAL THEOREMS AND THE TAYLOR POLYNOMIALS. QUALITATIVE STUDY OF THE GRAPH OF A FUNCTION.

UNIT HOURS 6:RIEMANN INTEGRAL – 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.

UNIT HOURS 10:INDEFINITE INTEGRAL – PRIMITIVE FUNCTIONS. THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. INDEFINITE INTEGRAL AND ITS PROPERTIES. INTEGRATION METHODS: DECOMPOSITION IN SUMS, INTEGRATION BY PARTS AND SUBSTITUTION.

UNIT HOURS 15: NUMERICAL SERIES– DEFINITIONS, EXAMPLES AND PRELIMINARY RESULTS. SERIES OF POSITIVE AND NEGATIVE TERMS. GEOMETRIC AND HARMONIC SERIES. TESTS FOR CONVERGENCE AND DIVERGENCE. ALTERNATING SERIES AND LEIBNIZ THEOREM. ABSOLUTELY CONVERGENT SERIES.

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 18 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 STUDENT WILL TAKE THE MINIMUM GRADE AT THE ORAL EXAMINATION (18/30) WHEN HE JUST KNOWS DEFINITIONS AND STATEMENTS OF THE ARGUMENTS IN THE PROGRAME OF THE COURSE BUT HE IS NOT ABLE TO CONNECT ARGUMENTS IN A CRITICAL WAY AS WELL AS TO GIVE CORRECT PROOFS OF THE MAIN RESULTS.
THE STUDENT WILL TAKE THE MAXIMUM GRADE AT THE ORAL EXAMINATION (30/30) WHEN HE KNOWS ALL THE ARGUMENTS OF THE COURSE AND HE IS ABLE TO CONNECT ARGUMENTS IN A CRITICAL WAY, WITH A PRECISE MATHEMATICAL LANGUAGE.
THE MAXIMUM GRADE WILL BE CUM LAUDE WHENEVER THE STUDENT SHOWS ABILITY IN CONNECTIONS BETWEEN DIFFERENT ARGUMENTS AND THE KEY INGRADIENTS OF THEIR PROOFS.
Texts
W. RUDIN: PRINCIPLES OF MATHEMATICAL ANALYSIS, MCGRAW-HILL, 1976, 342 PAGINE, ISBN 00-708-5613-3

OR

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 THE CONTAINS 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.