# Ingegneria Edile-Architettura | CALCULUS I

## Ingegneria Edile-Architettura CALCULUS I

 0660100001 DEPARTMENT OF CIVIL ENGINEERING EQF7 BUILDING ENGINEERING - ARCHITECTURE 2020/2021

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 60 LESSONS BASIC COMPULSORY SUBJECTS
 ANTONIO VITOLO T ALESSANDRA MEOLI
Objectives
LEARNING OUTCOMES: THE AIM OF THE COURSE IS THE STUDENT LEARNING ABOUT THE BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND CALCULUS, THE GEOMETRIC INTERPRETATION AND PHYSICAL APPLICATIONS.
LEARNING OUTCOMES
1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FINALIZED TO PROVIDE THE STUDENTS WITH THE MATHEMATICAL LANGUAGE, THE BASIC MATHEMATICAL NOTIONS AND THE GRAPHICAL REPRESENTATION EXSPECIALLY ABOUT THE FOLLOWING SUBJECTS: LIMITS, DIFFERENTIAL AND INTEGRAL CALCULUS, NUMERICAL SEQUENCES AND SERIES.
2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL BE ABLE TO FORMULATE MATHEMATICALLY AND TO SOLVE SIMPLE PROBLEMS OF THE APPLIED SCIENCES, EXPECIALLY IN THE CIVIL ENGINEERING FRAMEWORK. IN PARTICULAR, THE STUDENT WILL BE ABLE TO CALCULATE LIMITS, DERIVATIVES AND INTEGRALS, TO STUDY AND DRAW THE GRAPH OF A FUNCTION, TO COMPUTE AREAS, TO ESTABLISH THE CONVERGENCE OF A SERIES, MAKING COMPUTATIONS WITH COMPLEX NUMBERS.
3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO CHOOSE THE MOST APPROPRIATE MATHEMATICAL MODEL AND METHOD IN DIFFERENT SITUATIONS, TO CHECK THE VALIDITY OF A RESULT.
4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO EXPRESS WITH THE SUITABLE TECHNICAL LANGUAGE AND TO REPRESENT GRAPHICALLY THE LEARNED MATHEMATICAL NOTIONS AND TECHNIQUES, AND TO INTEGRATE THEM WITH THOSE ONES OF OTHER SCIENTIFIC DISCIPLINES.
Prerequisites
PREREQUISITES: SETS. REPRESENTATIONS OF REAL NUMBERS AND OPERATIONS. FIRST AND SECOND DEGREE EQUATIONS AND INEQUALITIES. TRIGONOMETRY. DECIMAL AND NATURAL LOGARITHM.

PROPAEDEUTIC EXAMS: 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. LINEARIZATION AND ERROR ESTIMATE (LECTURES/ EXERCISES: 9H/8H)
5. AREA PROBLEM. DEFINITE INTEGRAL. INTEGRALE MEAN. FUNDAMENTAL THEOREM OF CALCULUS. APPLICATIONS: AREAS, FORCES, WORK. 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).
THE ATTENDANCE IS MANDATORY. THE STUDENT IS REQUIRED TO ATTEND AT LEAST THE 70% OF THE COURSE, USING HIS PERSONAL BADGE.
Verification of learning
LEARNING ASSESSMENT WILL BE DONE THROUGH A WRITTEN AND ORAL EXAM AT THE END OF THE COURSE. MIDTERM WRITTEN PROOFS COULD BE DONE DURING 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, CALCULUS, A COMPLETE COURSE, 7TH EDITION, PEARSON.
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
LECTURE NOTES.