SERENA SPINA | CALCULUS I
SERENA SPINA CALCULUS I
cod. 0660100001
CALCULUS I
| 0660100001 | |
| DEPARTMENT OF CIVIL ENGINEERING | |
| EQF7 | |
| BUILDING ENGINEERING - ARCHITECTURE | |
| 2024/2025 |
| OBBLIGATORIO | |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2017 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/05 | 6 | 60 | LESSONS |
| Objectives | |
|---|---|
| GENERAL OBJECTIVE: LEARNING THE BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND CALCULATION FOR FUNCTIONS OF ONE VARIABLE, WITH ELEMENTS OF ANALYTICAL PLANE GEOMETRY. KNOWLEDGE AND UNDERSTANDING: ACQUISITION OF SKILLS RELATING TO BASIC MATHEMATICAL CONCEPTS AND THEIR GRAPHICAL REPRESENTATION WITH PARTICULAR REGARD TO THE FOLLOWING TOPICS: ANALYTICAL GEOMETRY, FUNCTIONS OF ONE VARIABLE, LIMITS, DIFFERENTIAL AND INTEGRAL CALCULUS, NUMERICAL SEQUENCES AND SERIES. ABILITY TO UNDERSTAND AND ACQUIRE MATHEMATICAL LANGUAGE. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: APPLICATION OF ACQUIRED KNOWLEDGE TO CALCULATE LIMITS, DERIVATIVES AND INTEGRALS; STUDY AND DRAW THE GRAPH OF A FUNCTION OF A VARIABLE IN THE PLANE; SOLVE MAXIMUM AND MINIMUM PROBLEMS; CALCULATE AREAS; CALCULATE THE LIMIT OF A SUCCESSION AND ESTABLISH THE CONVERGENCE OF A SEQUENCE; PERFORM CALCULATIONS WITH COMPLEX NUMBERS; DETERMINING THE CHARACTER OF A NUMERICAL SERIES. INDEPENDENT JUDGMENT: ABILITY TO CHOOSE THE MOST SUITABLE MATHEMATICAL MODELS AND METHODS FOR THE VARIOUS SITUATIONS AND VERIFY THE VALIDITY OF THE RESULTS OBTAINED FROM A QUALITATIVE AND QUANTITATIVE POINT OF VIEW. COMMUNICATION SKILLS: ABILITY TO EXPLAIN, WITH APPROPRIATE TECHNICAL LANGUAGE AND WITH ADEQUATE GRAPHICAL REPRESENTATION, THE ACQUIRED MATHEMATICAL NOTIONS AND METHODS, ALSO INTEGRATING THE ACQUIRED KNOWLEDGE WITH THAT TYPICAL OF OTHER DISCIPLINES. LEARNING ABILITY: CONSOLIDATION OF ACQUIRED KNOWLEDGE AND SKILLS TO LEARN WITHOUT DIFFICULTY MORE ADVANCED AND CONTENT MATHEMATICAL TOPICS THAN OTHER SCIENTIFIC DISCIPLINES THAT USE MATHEMATICAL TOOLS. |
| Prerequisites | |
|---|---|
| PREREQUISITES: SETS. REPRESENTATIONS OF REAL NUMBERS AND OPERATIONS. FIRST AND SECOND DEGREE EQUATIONS AND INEQUATIONS. TRIGONOMETRY. DECIMAL AND NATURAL LOGARITHM. PREPARATIONS: NONE |
| Contents | |
|---|---|
| THE COURSE IS STRUCTURED AS FOLLOWS: 1. PRELIMINARIES: SETS. REAL NUMBERS. REAL LINE. LINEAR, QUADRATIC AND RATIONAL EQUATIONS AND INEQUALITIES. CARTESIAN FRAME OF REFERENCE IN THE PLANE. LINES AND PARABOLAS (LECTURES/ EXERCISES: 3H/5H) 2. FUNCTIONS AND GRAPHS: DEFINITIONS AND PROPERTIES. MONOTONICITY. ELEMENTARY FUNCTIONS: POWERS WITH INTEGER AND FRACTIONAL EXPONENTS, SINUS, COSINUS, EXPONENTIAL, LOGARITHM. COMPOSITE AND INVERSE FUNCTIONS. IRRATIONAL AND TRANSCENDENTAL EQUATIONS (LECTURES/ EXERCISES: 3H/3H) 3. LIMITS AND CONTINUITY: DEFINITIONS AND PROPERTIES. COMPARISON THEOREMS. DISCONTINUITY. ESTREME VALUE THEOREM AND INTERMEDIATE VALUE THEOREM. APPROXIMATE CALCULUS OF SOLUTIONS OF EQUATIONS (LECTURES/ EXERCISES: 3H/3H) 4. DIFFERENTIABILITY: DEFINITIONS AND PROPERTIES. DERIVATIVE AND TANGENT LINE. SPEED AND ACCELERATION. COMPUTATION RULES. DIFFERENTIALS. INDEFINITE INTEGRATION. LAGRANGE MEAN VALUE THEOREM. APPLICATIONS: MONOTONICITY, MAXIMA AND MINIMA. HIGHER ORDER DERIVATIVES. TAYLOR FORMULA. CONVEXITY, CONCAVITY AND INLECTION POINTS. GRAPH OF A FUNCTION. (LECTURES/ EXERCISES: 9H/8H) 5. AREA PROBLEM. DEFINITE INTEGRAL. INTEGRALE MEAN. FUNDAMENTAL THEOREM OF CALCULUS. INTEGRATION TECHNIQUES. COMPLEX NUMBERS. IMPROPER INTEGRALS (LECTURES/ EXERCISES: 9H/8H) 6. NUMERICAL SEQUENCES AND SERIE: CONVERGENCE OF A SEQUENCE. MONOTONE SEQUENCES. CONVERGENCE AND SUM OF A SERIES. GEOMETRIC AND EXPONENTIAL SERIES. CONVERGENCE CRITERIA (LECTURES/ EXERCISES: 3H/3H) |
| Teaching Methods | |
|---|---|
| THE COURSE CONSISTS OF 30 HOURS OF THEORETIC FRONTAL LECTURES (3 ECTS) WITH EXAMPLES AND 30 HOURS OF EXCERCISE SESSIONS (3 ECTS), IN TOTAL 60 HOURS (6 ECTS). COURSE ATTENDANCE IS COMPULSORY: THE MINIMUM PERCENTAGE OF ATTENDANCE REQUIRED TO TAKE THE EXAM IS 70% (AS PER CDS TEACHING REGULATIONS). THE LECTURER WILL VERIFY THE ACHIEVEMENT OF THE REQUIRED PERCENTAGE OF ATTENDANCE BOTH THROUGH ELECTRONIC RECORDING AND THROUGH VERIFICATION OF THE STUDENT'S ACTUAL PARTICIPATION IN CLASS. |
| Verification of learning | |
|---|---|
| LEARNING ASSESSMENT WILL BE DONE THROUGH A WRITTEN AND ORAL EXAM AT THE END OF THE COURSE. THE WRITTEN EXAM CONSISTS OF EXERCISES OR PROBLEMS FOR EVALUATING THE KNOWLEDGE APPLICATION CAPABILITY RELATED TO DIFFERENTIAL AND INTEGRAL CALCULUS OF FUNCTIONS OF ONE VARIABLE, NUMERICAL SERIES AND COMPLEX NUMBERS. THE ORAL EXAM, WHICH MAY INCLUDE EXERCISES, CONSISTS OF QUESTIONS ON THE SAME SUBJECTS AND SERVES TO EVALUATE THE LEVEL OF STUDENT’S THEORETICAL KNOWLEDGE, MAKING JUDGEMENT AND COMMUNICATION SKILLS. TO ACCESS THE ORAL EXAM, THE GRADE OF THE WRITTEN PROOF HAS TO BE NOT LESS THAN 18/30. THE FINAL GRADE WILL BE EXPRESSED IN THIRTIES. IT WILL BE NORMALLY THE MEAN OF PARTIAL EVALUATIONS. THE MINIMUM GRADE (18) CORRESPONDS TO A FRAGMENTARY THEORETICAL KNOWLEDGE AND A LIMITED CAPABILITY TO USE IT IN THE APPLICATIONS. THE MAXIMUM GRADE (30) CORRESPONDS TO A COMPLETE KNOWLEDGE OF THEORETICAL CONTENTS AND METHODOLOGIES, A CONSIDERABLE CAPABILITY TO USE IT IN THE APPLICATIONS AND COMMUNICATION SKILLS. HONORS CAN BE OBTAINED BY A STUDENT WHO EXHIBITS A NOTEWORTHY THEORETICAL KNOWLEDGE, A PERFECT COMMAND OF SCIENTIFIC LANGUAGE AND HIGH DEGREE OF AUTONOMY ALSO IN NEW CONTEXTS. |
| Texts | |
|---|---|
| ROBERT A. ADAMS, CRISTOPHER ESSEX, CALCOLO DIFFERENZIALE 1, 5A EDIZIONE, CASA EDITRICE AMBROSIANA. P.MARCELLINI, C.SBORDONE, ESERCITAZIONI DI MATEMATICA, VOLUME 1, PARTE I, LIGUORI EDITORE P.MARCELLINI, C.SBORDONE, ESERCITAZIONI DI MATEMATICA, VOLUME 1, PARTE II, LIGUORI EDITORE APPUNTI DELLE LEZIONI |
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