# Computer science | MATHEMATICAL ANALYSIS

## Computer science MATHEMATICAL ANALYSIS

 0512100001 COMPUTER SCIENCE EQF6 COMPUTER SCIENCE 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 SPRING SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 48 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS MAT/05 3 24 EXERCISES SUPPLEMENTARY COMPULSORY SUBJECTS

 ELMO BENEDETTO T
ExamDate
APPELLO PROF. BENEDETTO13/06/2023 - 12:00
Objectives
KNOWLEDGE AND UNDERSTANDING: TO PROVIDE STUDENTS WITH THE BASIC NOTIONS OF CALCULUS.
APPLYING KNOWLEDGE AND UNDERSTANDING: LEARN CALCULUS THEORY AND APPLICATIONS, SO AS TO BE ABLE TO SOLVE A RANGE OF SIMPLE EXERCISES. IN PARTICULAR, STUDENTS WILL BE ABLE TO DRAW THE GRAPHIC OF A FUNCTION GIVEN ITS ALGEBRAIC FORM AND TO CALCULATE SIMPLE INTEGRALS. TO BE ABLE TO PRESENT, IN A CLEAR AND RIGOROUS MANNER, THE THEOREMS, DEFINITIONS AND PROBLEMS STUDIED THROUGHOUT THE COURSE. TO BE ABLE TO USE THE ACQUIRED KNOWLEDGE IN REASONING AND ALGORITHMS
Prerequisites
INTRODUCTIVE ELEMENTAL NOTIONS, EQUATIONS AND INEQUALITIES OF FIRST AND SECOND ORDER, ALGEBRAIC, FRACTIONAL, EXPONENTIAL AND LOGARITHMIC.
PREPARATORY TEACHINGS
NONE
Contents
NUMERICAL SETS.
INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS OF A SET. INTRODUCTION TO REAL NUMBERS. EXTREMES OF A NUMERICAL SET. INTERVALS OF R. NEIGHBORHOODS, POINTS OF ACCUMULATION. CLOSED SETS AND OPEN SETS.
(4,1)

COMBINATORIAL CALCULUS AND AND PRINCIPLE OF INDUCTION
(2,1)

COMPLEX NUMBERS
(3,2)

REAL FUNCTIONS.
DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF A FUNCTION. EXTREMES OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSITE FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND N-TH ROOT FUNCTIONS, EXPONENTIAL, LOGARITHMIC FUNCTION, POWER FUNCTION.
(6,1)

BASIC NOTIONS OF EQUATIONS AND INEQUALITIES.
EQUATIONS OF FIRST ORDER. QUADRATIC EQUATIONS. IRRATIONAL EQUATIONS. EXPONENTIAL AND LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. LINEAR INEQUALITIES. INEQUALITIES OF THE SECOND ORDER. FACTIONAL INEQUALITIES. IRRATIONAL INEQUALITIES. EXPONENTIAL AND LOGARITHMIC INEQUALITIES. SYSTEMS OF INEQUALITIES.
(1,3)

NUMERICAL SEQUENCES (BASIS ELEMENTS).

DEFINITION. NUMERICAL SEQUENCES. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES.
(3,1)

LIMITS OF A FUNCTION.
DEFINITION. RIGHT AND LEFT HAND-SIDE LIMITS. UNIQUENESS THEOREM. COMPARISON THEOREMS. OPERATIONS AND INDETERMINATE FORMS. KNOWN LIMITS.
(5,3)

CONTINUOUS FUNCTIONS.

DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM. ZEROS THEOREM.
(2,0)

DERIVATIVE OF A FUNCTION.

DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING, THE TANGENT LINE TO THE GRAPH OF A FUNCTION. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY FUNCTIONS. DERIVATIVES OF COMPOSITE FUNCTION. HIGHER ORDER DERIVATIVES.
(5,2)

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS.

ROLLE'S THEOREM. CAUCHY'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. THEOREM OF DE L'HOSPITAL. CONDITIONS FOR MAXIMA AND MINIMA.
(5,2)

GRAPH OF A FUNCTION.

ASYMPTOTES OF A GRAPH. SEARCH OF MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. GRAPH OF A FUNCTION BY ITS CHARACTERISTIC ELEMENTS.
(3,3)

INTEGRATION OF ONE VARIABLE FUNCTIONS.

DEFINITION OF INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS.
(3,3)

MULTIVARIABLE FUNCTIONS.

DEFINITIONS. LIMITS AND CONTINUITY. WEIERSTRASS THEOREM. PARTIAL DIFFERENTIATION. SCHWARZ THEOREM. GRADIENT AND DIFFERENTIABILITY. DIRECTIONAL DERIVATIVES. LOCAL MINIMA AND MAXIMA.
(2,1)

TAYLOR'S FORMULA
(2,0)

NUMERICAL SERIES: DEFINITIONS AND PROPERTIES
(2,1)

TOTAL HOURS: (48,24)
Teaching Methods
FRONTAL LECTURES, STUDY GROPUS, EXERCISES, INTERACTION WITH THE PROFESSOR IN THE CLASSROOM AND THROUGH E-MAIL.
Verification of learning for the student group
Verification of learning
THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE:
-THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE TEACHING.
-THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL PROOFS
• THE SKILL OF PROVING THEOREMS
• THE SKILL OF SOLVING EXERCISES
• THE SKILL TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHOD IN EXERCISES SOLVING
• THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE TEACHING.
THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW.
WRITTEN PROOF: THE WRITTEN PROOF CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE LESSONS. THE PROOF WILL LAST 2 HOURS.
ORAL INTERVIEW: THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS, DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.
Texts
P.DIGIRONIMO-G.IOVANE-E.BENEDETTO-A.BRISCIONE, ANALISI MATEMATICA PER INFORMATICI-TEORIA ED ESERCIZI, ED. ARACNE