Tiziana DURANTE | CALCULUS 1
Tiziana DURANTE CALCULUS 1
cod. 0612800001
CALCULUS 1
| 0612800001 | |
| DEPARTMENT OF INFORMATION AND ELECTRICAL ENGINEERING AND APPLIED MATHEMATICS | |
| EQF6 | |
| INFORMATION ENGINEERING FOR DIGITAL MEDICINE | |
| 2024/2025 |
| OBBLIGATORIO | |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2022 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/05 | 4 | 32 | LESSONS | |
| MAT/05 | 5 | 40 | EXERCISES |
| Objectives | |
|---|---|
| THE COURSE PRESENTS THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE TRAINING OBJECTIVES THE TEACHING CONSISTS IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS IN THE ABILITY TO USE THE RELATED CALCULATION TOOLS. KNOWLEDGE AND UNDERSTANDING NUMERICAL SETS. REAL FUNCTIONS. REFERENCES ON EQUATIONS AND INEQUATIONS. NUMERICAL SUCCESSIONS. LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS. DERIVATIVE OF A FUNCTION. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULATION. STUDY OF THE GRAPH OF A FUNCTION. INTEGRATION OF FUNCTIONS OF A VARIABLE. SOFTWARE TOOLS FOR MATHEMATICS. APPLYING KNOWLEDGE AND UNDERSTANDING APPLY THE THEOREMS AND RULES STUDIED TO TROUBLESHOOTING. PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES, INTEGRAL. CONDUCT THE STUDY OF THE GRAPH OF A FUNCTION. |
| Prerequisites | |
|---|---|
| IN ORDER TO REACH THE OBJECTIVES STUDENTS SHOULD HAVE A BACKGROUND RELATING TO ALGEBRA, IN PARTICULAR ALGEBRAIC EQUATIONS AND INEQUALITIES, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL INEQUALITIES, AND THEY SHOULD UNDERSTAND TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC THEORY OF TRIGONOMETRIC FUNCTIONS. |
| Contents | |
|---|---|
| TEACHING UNIT 1: NUMERICAL SETS. (LECTURE/ PRACTICE/LABORATORY HOURS 2/0/0) - 1 (2 HOURS LECTURES): INTRODUCTION TO THE THEORY OF THE SETS. INTRODUCTION TO REAL NUMBERS. EXTREME OF A NUMERICAL SET, INTERVALS, NEIGHBOURHOODS, ACCUMULATION POINTS, CLOSED AND OPEN SETS. INDUCTION PRINCIPLE. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO ACQUIRE THE CONCEPT OF SET, EXTREMES, INTERVALS, NEIGHBOURHOODS, ACCUMULATION POINTS, CLOSED AND OPEN SETS. APPLYING KNOWLEDGE AND UNDERSTANDING: TEACHING UNIT 2: COMPLEX NUMBERS. (LECTURE/ PRACTICE/LABORATORY HOURS 2/2/0) - 2 (2 HOURS LECTURE): IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. ALGEBRAIC FORM AND TRIGONOMETRIC FORM. DE MOIVRE FORMULA. N-TH ROOTS. - 3 (2 HOURS PRACTICE): EXAMPLES OF REPRESENTATION OF A COMPLEX NUMBER IN ALGEBRAIC AND TRIGONOMETRIC FORM, CALCULUS OF SUMS, PRODUCTS, RATIO, POWERS AND N-TH ROOTS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: KNOWLEDGE OF THE CONCEPT OF COMPLEX NUMBER. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO REPRESENT A COMPLEX NUMBER IN ALGEBRAIC AND TRIGONOMETRIC FORM, TO CALCULATE SUMS, PRODUCTS, RATIO, POWERS AND N-TH ROOTS. TEACHING UNIT 3: REAL FUNCTIONS. (LECTURE/ PRACTICE/LABORATORY HOURS 4/4/0) - 4 (2 HOURS LECTURE): DEFINITION. EXTREME OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSED FUNCTIONS. INVERTIBLE FUNCTIONS. NUMERICAL SEQUENCES. - 5 (2 HOURS LECTURE): ELEMENTARY FUNCTIONS: N-TH POWER FUNCTION AND N-TH ROOT, EXPONENTIAL FUNCTION, LOGARITHMIC FUNCTION, POWER FUNCTION, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES. - 6 (2 HOURS PRACTICE): BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, SECOND ORDER, IRRATIONAL, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. FIRST ORDER, SECOND ORDER, RATIONAL, IRRATIONAL, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS. - 7 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF DOMAINS OF FUNCTIONS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: KNOWLEDGE OF THE CONCEPT OF FUNCTION. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO CALCULATE THE DOMAIN OF A FUNCTION. TEACHING UNIT 4: LIMITS. (LECTURE/ PRACTICE/LABORATORY HOURS 4/6/0) - 8 (2 HOURS LECTURE): LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. CONVERGENT, DIVERGENT AND OSCILLATING NUMERICAL SEQUENCES. MONOTONE SEQUENCES. NEPERO’S NUMBER. - 9 (2 HOURS LECTURE): UNIQUENESS AND COMPARISON THEOREMS. -10 (2 HOURS PRACTICE): OPERATIONS WITH LIMITS AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. - 11 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF LIMITS. - 12 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF LIMITS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: KNOWLEDGE OF THE CONCEPT OF LIMIT. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO CALCULATE LIMITS. TEACHING UNIT 5: NUMERICAL SERIES. (LECTURE/ PRACTICE/LABORATORY HOURS 4/2/0) - 13 (2 HOURS LECTURE): DEFINITIONS. CONVERGENT, DIVERGENT AND OSCILLATING SERIES. HARMONIC AND GEOMETRIC SERIES. POSITIVE AND NON NEGATIVE SERIES AND CONVERGENCE CRITERIA: CRITERIA OF COMPARISON, OF THE RATIO, OF THE ROOT. - 14 (2 HOURS LECTURE): SERIES WITH ALTERNATE TERMS. LEIBNITZ CRITERION. ABSOLUTE CONVERGENCE. - 15 (2 HOURS PRACTICE): EXAMPLES OF STUDIES OF THE NATURE OF POSITIVE, NON NEGATIVE AND ALTERNATING SERIES. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO ACQUIRE THE CONCEPT OF CONVERGENT, DIVERGENT AND OSCILLATING SERIES. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO STUDY THE NATURE OF NATURE OF POSITIVE, NON NEGATIVE AND ALTERNATING SERIES. TEACHING UNIT 6: CONTINUITY. (LECTURE/ PRACTICE/LABORATORY HOURS 2/2/0) - 16 (2 HOURS LECTURE): CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS.. .-17 (2 HOURS PRACTICE): EXAMPLES OF DISCONTINUITIES POINTS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: KNOWLEDGE OF CONCEPT OF CONTINUITY. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO CLASSIFY DISCONTINUITY POINTS. TEACHING UNIT 7: DERIVATIVES. (LECTURE/ PRACTICE/LABORATORY HOURS 10/4/0) - 18 (2 HOURS LECTURE): DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. TANGENTIAL LINE. DIFFERENTIABILITY AND CONTINUITY. DISCONTINUITIES OF THE DERIVATIVE. - 19 (2 HOURS LECTURE): DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION. - 20 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF DERIVATIVES. - 21 (2 HOURS LECTURE): THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. - 22 (2 HOURS LECTURE): CONDITIONS FOR RELATIVE MAXIMA AND MINIMA. CONCAVITY, CONVEXITY, INFLECTION POINTS. - 23 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF RELATIVE AND ABSOLUTE MAXIMA AND MINIMA. - 24 (2 HOURS LECTURE): TAYLOR AND MAC-LAURIN FORMULAS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: KNOWLEDGE OF DERIVATIVE AND ITS GEOMETRICAL MEANING. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO CALCULATE DERIVATIVES, CLASSIFY DISCONTINUITY POINTS OF THE DERIVATIVE, CALCULATE RELATIVE AND ABSOLUTE MINIMA AND MAXIMA. TEACHING UNIT 8: GRAPH OF A FUNCTION. (LECTURE/ PRACTICE/LABORATORY HOURS 2/8/0) - 25 (2 HOURS LECTURE): GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS, INFLECTION POINTS. - 26 (2 HOURS PRACTICE): EXAMPLES OF GRAPH OF FUNCTION USING ITS CHARACTERISTIC ELEMENTS. - 27 (2 HOURS PRACTICE): EXAMPLES OF GRAPH OF FUNCTION USING ITS CHARACTERISTIC ELEMENTS -28 (2 HOURS PRACTICE): EXAMPLES OF GRAPH OF FUNCTION USING ITS CHARACTERISTIC ELEMENTS - 29 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF ABSOLUTE MAXIMA AND MINIMA. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO ACQUIRE THE CHARACTERISTIC ELEMENTS OF A FUNCTION. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO DRAW THE GRAPH OF A FUNCTION. TEACHING UNIT 9: INDEFINITE AND DEFINITE INTEGRALS. (LECTURE/ PRACTICE/LABORATORY HOURS 4/6/0) - 30 (2 HOURS LECTURE): INTEGRATION OF ONE VARIABLE FUNCTIONS: PRIMITIVE FUNCTIONS AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. INTEGRATION RULES AND METHODS. INTEGRATION BY PARTS AND BY SUBSTITUTION. - 31 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF IMMEDIATE INTEGRALS. - 32 (2 HOURS PRACTICE): EXAMPLES OF CALCULUS OF INTEGRALS BY PARTS AND SUBSTITUTION - 33 (2 HOURS PRACTICE): EXAMPLES OF INTEGRATION OF RATIONAL FUNCTIONS. - 34 (2 ORE LECTURE) DEFINITE INTEGRAL AND GEOMETRICAL MEANING. MEAN VALUE THEOREM. FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. KNOWLEDGE AND UNDERSTANDING CAPABILITIES: TO ACQUIRE THE CONCEPT OF INDEFINITE AND DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO APPLY DIFFERENT INTEGRATION RULES AND METHODS. TEACHING UNIT 10: MATH SOFTWARE TOOLS. (LECTURE/ PRACTICE/LABORATORY HOURS 0/4/0) - 35 (2 HOURS PRACTICE): BUILT-IN FUNCTIONS TO COMPUTE OPERATIONS WITH COMPLEX NUMBERS, LIMITS, DERIVATIVES AND INTEGRALS. - 36 (2 HOURS PRACTICE): BUILT-IN FUNCTIONS TO STUDY THE GRAPH OF A FUNCTION AND THE NATURE OF A NUMERICAL SERIES: APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO SOLVE THE EXERCISES COVERED DURING THE COURSE WITH A MATHEMATICAL SOFTWARE. TOTAL LECTURE/ PRACTICE/LABORATORY HOURS 34/38/0 |
| Teaching Methods | |
|---|---|
| THE COURSE CONSISTS IN THEORETICAL LECTURES AND EXERCISES. 72 HOURS IN TOTAL: 38 HOURS FOR THE THEORY AND 34 HOURS FOR THE EXERCISES. THE COURSE IS DELIVERED IN PRESENCE WITH MANDATORY FREQUENCY, CERTIFIED BY THE STUDENT THROUGH THE USE OF THE PERSONAL BADGE. IN ORDER TO BE ABLE TO SUSTAIN THE FINAL VERIFICATION OF PROFITS AND OBTAIN THE CFU RELATED TO THE EDUCATIONAL ACTIVITY, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE SCHEDULED HOURS OF TEACHING ACTIVITY. |
| Verification of learning | |
|---|---|
| THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE. THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL TEST. THE SKILL OF PROVING THEOREMS. THE SKILL OF SOLVING EXERCISES. THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING. THE ABILITY TO USE THE ACQUIRED KNOWLEDGE. THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL EXAMINATION. WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE (THERE ARE SAMPLES AVAILABLE ON THE WEBSITE) AND THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. THERE WILL BE A MID TERM TEST CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENTS ON THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS. THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS, AS PROOFS OF THEOREMS AND IN SOLVING EXERCISES. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL VALUTATION OF THE STUDENT. |
| Texts | |
|---|---|
| BASIC TEXT FOR THEORY: P.MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO, LIGUORI EDITORE. BASIC TEXT FOR THE EXERCISES: P.MARCELLINI-C.SBORDONE}, ESERCITAZIONI DI MATEMATICA I, VOL.I, PARTE I,II, LIGUORI EDITORE. SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS. |
| More Information | |
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| THE COURSE IS HELD IN ITALIAN |
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