Computer science | MATHEMATICAL ANALYSIS
Computer science MATHEMATICAL ANALYSIS
cod. 0512100001
MATHEMATICAL ANALYSIS
| 0512100001 | |
| DIPARTIMENTO DI INFORMATICA | |
| EQF6 | |
| COMPUTER SCIENCE | |
| 2016/2017 |
| OBBLIGATORIO | |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2015 | |
| SECONDO SEMESTRE |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/05 | 6 | 48 | LESSONS | |
| MAT/05 | 3 | 24 | EXERCISES |
| Objectives | |
|---|---|
| KNOWLEDGE AND UNDERSTANDING: TO PROVIDE STUDENTS WITH THE BASIC NOTIONS OF CALCULUS: 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. APPLYING KNOWLEDGE AND UNDERSTANDING: LEARN CALCULUS THEORY AND APPLICATIONS. TO BE ABLE TO PRESENT, IN A CLEAR AND RIGOROUS MANNER, THE THEOREMS, DEFINITIONS AND PROBLEMS STUDIED THROUGHOUT THE COURSE. |
| Prerequisites | |
|---|---|
| ELEMENTARY ASPECTS OF ALGEBRA ARE MANDATORY AS WELL AS FAMILIARITY WITH SOLUTION METHODS FOR FIRST AND SECOND ORDER EQUALITIES AND INEQUALITIES. KNOWLEDGE OF SOME ELEMENTS OF TRIGONOMETRY ARE ALSO CONSIDERED A PREREQUISITE. |
| Contents | |
|---|---|
| REAL NUMBERS THE AXIOMS OF REAL NUMBERS AND THEIR CONSEQUENCES. NATURAL NUMBERS, INTEGERS AND RATIONAL NUMBERS. NUMERICAL SETS: BOUNDED SETS, MAXIMUM AND MINIMUM, SUPREMUM AND INFIMUM. UNIQUENESS OF THE MINIMUM (WITH PROOF). EXISTENCE THEOREM FOR THE SUPREMUM OF A NUMERICAL SET (WITH PROOF). UNBOUNDED SETS. INTERVALS. ARCHIMEDEAN PROPERTY (WITH PROOF). NON-COMPLETENESS OF THE RATIONAL NUMBERS (WITH PROOF). REAL FUNCTIONS OF A REAL VARIABLE DEFINITION AND NOTATIONS. EXTREMA. CARTESIAN REPRESENTATION. ODD FUNCTIONS, EVEN FUNCTIONS AND PERIODIC FUNCTIONS. COMPOSITION OF FUNCTIONS. INVERSE FUNCTIONS. MONOTONE FUNCTIONS AND THEIR INVERSE (WITH PROOF) . ELEMENTARY FUNCTIONS LINEAR FUNCTIONS. THE ABSOLUTE VALUE. TRIANGUAR INEQUALITY. POWER FUNCTIONS. LOGARITHM AND EXPONENTIAL FUNCTIONS. TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS. NUMERICAL SEQUENCES DEFINITION AND NOTATIONS. CONVERGING SEQUENCES: DEFINITION AND EXAMPLES. UNIQUENESS OF THE LIMIT (WITH PROOF). DIVERGENT SEQUENCES: DEFINITIONS AND EXAMPLES. IRREGULAR SEQUENCES. BOUNDEDNESS OF CONVERGENT SEQUENCES (WITH PROOF). OPERATIONS WITH CONVERGING SEQUENCES (PROOFS OF THE SUM AND THE PRODUCT). THEOREM OF THE PERMANENCE OF THE SIGN AND COROLLARIES (WITH PROOFS). THEOREM OF THE “CARABINIERI” (WITH PROOF). THEOREM OF THE PRODUCT OF A BOUNDED SEQUENCE TIMES AN INFINITESIMAL ONE. MONOTONE SEQUENCES AND THEIR THEOREM (WITH PROOF). THE NUMBER E. NOTABLE SPECIAL LIMITS. EXTRACTED SEQUENCES. THEOREM OF BOLZANO - WEIERSTRASS . LIMITS OF FUNCTIONS DEFINITION AND NOTATIONS. LINK BETWEEN LIMITS OF FUNCTIONS AND LIMITS OF SEQUENCES . UNIQUENESS OF THE LIMIT. OPERATIONS WITH LIMITS. INDETERMINATE FORMS. LIMITS OF COMPOSITE FUNCTIONS. LIMITS OF THE ELEMENTARY FUNCTIONS AT THE EXTREMA OF THEIR DOMAIN. NOTABLE SPECIAL LIMITS (WITH PROOFS). CONTINUOUS FUNCTIONS DEFINITION. CONTINUITY OF ELEMENTARY FUNCTIONS. OPERATIONS WITH CONTINUOUS FUNCTIONS. COMPOSITION OF CONTINUOUS FUNCTIONS. SIGN PERMANENCE THEOREM (WITH PROOF). MEAN VALUE THEOREM. WEIERSTRASS EXISTENCE THEOREM. DERIVATIVES DEFINITION AND GEOMETRICAL MEANING. CONTINUITY OF DIFFERENTIABLE FUNCTIONS (WITH PROOF). OPERATIONS WITH DERIVATIVES (WITH PROOFS). DERIVATIVE OF A COMPOSITION OF FUNCTION. DERIVATIVE OF THE INVERSE FUNCTION. DERIVATIVES OF THE ELEMENTARY FUNCTIONS (WITH PROOFS). HIGHER ORDER DERIVATIVES. THEOREMS OF DE L'HOPITAL. TAYLOR'S FORMULA . APPLICATIONS OF DERIVATIVES. STUDY OF REAL FUNCTIONS OF A REAL VARIABLE MAXIMA AND MINIMA OF A REAL FUNCTION OF ONE REAL VARIABLE. FERMAT'S THEOREM (WITH PROOF). ROLLE'S THEOREM (WITH PROOF). LAGRANGE'S THEOREM (GRAPHICAL INTERPRETATION). MONOTONICITY CRITERION (WITH PROOF). STRICT MONOTONICITY CRITERION. CHARACTERIZATION OF CONSTANT FUNCTIONS IN AN INTERVAL (WITH PROOF). CONVEX AND CONCAVE FUNCTIONS. CRITERION OF CONVEXITY. SUFFICIENT CONDITIONS FOR THE DETERMINATION OF THE RELATIVE EXTREMA OF A FUNCTION. STUDY OF A FUNCTION’S GRAPH. RIEMANN INTEGRATION DEFINITION IN THE CASE OF A BOUNDED FUNCTION AND NOTATIONS. GEOMETRIC INTERPRETATION. ADDITIVITY AND LINEARITY PROPERTIES. INTEGRABILITY OF CONTINUOUS FUNCTIONS. THEOREMS OF THE INTEGRAL MEAN VALUES (WITH PROOFS) . INDEFINITE INTEGRALS FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS (WITH PROOF). CHARACTERIZATION OF THE PRIMITIVE OF A FUNCTION IN AN INTERVAL (WITH PROOF). FUNDAMENTAL FORMULA OF THE INTEGRAL CALCULUS (WITH PROOF). INDEFINITE INTEGRATION AND ITS PROPERTIES. INTEGRATION BY SUM DECOMPOSITION. INTEGRATION OF RATIONAL FUNCTIONS. INTEGRATION BY PARTS. SOME TYPES OF INTEGRATION BY SUBSTITUTION. |
| Teaching Methods | |
|---|---|
| • FRONTAL LESSON • PRACTICE LESSON |
| Verification of learning | |
|---|---|
| THE KNOWLEDGE AND UNDERSTANDING OF THE TOPICS DESCRIBED WITHIN THE COURSE WILL BE TESTED BY MEANS OF A FINAL WRITTEN EXAMINATION CONTAINING THEORY AND EXERCISES TO TEST THE KNOWLEDGE OF THE BASIC NOTIONS OF CALCULUS (TO BE ABLE TO DRAW THE GRAPHIC OF A FUNCTION GIVEN ITS ALGEBRAIC FORM AND TO CALCULATE SIMPLE INTEGRALS). THE WRITTEN EXAMINATION WILL BE FOLLOWED BY AN ORAL EXAM TO VERIFY THAT THE STUDENT IS ABLE TO PRESENT, IN A CLEAR AND RIGOROUS MANNER, THE THEOREMS, DEFINITIONS AND PROBLEMS STUDIED THROUGHOUT THE COURSE. |
| Texts | |
|---|---|
| TESTI ADOTTATI • P. MARCELLINI - C. SBORDONE, “ANALISI MATEMATICA UNO “, LIGUORI EDITORE • P. MARCELLINI - C. SBORDONE, “ELEMENTI DI ANALISI MATEMATICA UNO “, LIGUORI EDITORE TESTI DI APPROFONDIMENTO • M. TROISI “ANALISI MATEMATICA I“, LIGUORI EDITORE • M.BRAMANTI-C.PAGANI-S. SALSA, “ANALISI MATEMATICA I“, LIGUORI ZANICHELLI • P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA I “, LIGUORI EDITORE • A. ALVINO - L. CARBONE- G. TROMBETTI, “ESERCITAZIONI DI MATEMATICA I “, LIGUORI EDITORE |
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