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MATHEMATICAL ANALYSIS 1

CODE 52474
ACADEMIC YEAR 2022/2023
CREDITS
  • 12 cfu during the 1st year of 8758 FISICA (L-30) - GENOVA
  • 16 cfu during the 1st year of 8760 MATEMATICA (L-35) - GENOVA
  • 16 cfu during the 1st year of 8766 STATISTICA MATEM. E TRATTAM. INFORMATICO DEI DATI (L-35) - GENOVA
  • SCIENTIFIC DISCIPLINARY SECTOR MAT/05
    LANGUAGE Italian
    TEACHING LOCATION
  • GENOVA
  • PREREQUISITES
    Prerequisites (for future units)
    This unit is a prerequisite for:
    • PHYSICS 8758 (coorte 2022/2023)
    • PHYSICS II 57049
    • ANALYTICAL MECHANICS 25911
    • GENERAL PHYSICS 3 57050
    • MATHEMATICAL ANALYSIS 2 57048
    MODULES This unit is composed by:
    TEACHING MATERIALS AULAWEB

    OVERVIEW

    Language: Italian

     

    AIMS AND CONTENT

    LEARNING OUTCOMES

    Rigorous treatment of Mathematical Analysis, focusing on differential and integral calculus of functions of one real variable.

    PREREQUISITES

    Elementary algebra; trigonometry

    TEACHING METHODS

    Traditional: blackboard.

    SYLLABUS/CONTENT

     1. Real numbers. The axioms of ordered fields. Absolute value. Natural and integer numbers. Rational numbers and their geometric representation. Completeness and its consequences. Real numbers and the straight line. Archimedean property. Decimal representations.

    2. Functions. Relations, functions, domain, codomain, image and graph of a function. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Trigonometric functions. The exponential function on rational numbers.

    3. Limits. Metric and e topological properties of R. Continuity. Operations with continuous functions. Limits and their properties. The algebra of limits. Comparison theorems. Limits of monotone functions. Limits of compositions and change of variables. Sequences and their limits. Sunsequences.  Bolzano-Weierstrass' theorem. Cauchy sequences. Sequences defined by recurrence and their limits. Neper's number e.

    4. Global properties of continuous functions. Weierstrass' theorem. Zeroes of continuous functions. Intermediate value theorem.  Continuity and monotonicity. Continuity of the inverse function. Uniform continuity. Heine-Cantor's theorem. The exponential funcion on real numbers.

    5. Differential calculus. The derivative: definition and elementary properties. Differentiability and the properties of the differential.  Derivative of compositions and inverse functions. Derivatives of elementary functions. Higher order derivatives. The classical theorems by Rolle, Lagrange and  Cauchy and their consequences. The theorem of de l'Hopital. Local comparison of functions. Vanishing and diverging functions. Taylor's formula. Convexity. Study of monotonicity and convexity by means of first and second derivatives.  Newton's method. Iterative procedures for the solution of equations.

    6.The  indefinite integral. Integration techniques.  Integration of elementary functions.  Integration by parts and by substitution.  Integration of rational functions. 

    7. The Riemann integral. Definition and properties of the definite integral. Integrability of continuous and monotonic functions. The oriented integral. The integral mean theorem. Relations between derivation and integration: integral functions, the fundamental theorem of calculus and its consequences. Improper integrals. Convergence criteria. 

    8. Series.  Geometric and telescopic series. Convergence. Series with non negative terms: comparison, root and ratio criteria; condensation, order and integral tests.  Alternating series and Leibniz' theorem.

    9. Differential equations. Separation of variables. Linear first order equations Second order linear equations with constant coefficients.

    RECOMMENDED READING/BIBLIOGRAPHY

    A.Bacciotti, F.Ricci - Analisi Matematica I - Liguori Editore

    M. Baronti, F. De Mari, R. van der Putten, I. Venturi - Calculus Problems, Springer, 2016

    Further readings will be posted on the web page (AULAWEB)

    TEACHERS AND EXAM BOARD

    LESSONS

    LESSONS START

    The class will start according to the academic calendar.

    Class schedule

    All class schedules are posted on the EasyAcademy portal.

    EXAMS

    EXAM DESCRIPTION

    The exam consists of a written test and an oral test.
    Students enrolled in the course of study in Physics are not required to study the proofs of the theorems that will be the object of learning during the second semester.

    ASSESSMENT METHODS

    1. Two intermediate written tests  will be provided during the year. If a student obtains an average mark greater than or equal to 18/30 and if he scores at least 15/30 in both, the average of the two marks counts as a written test. 
    
    2. A written test with a score greater than or equal to 12/30 allows access to the oral test.
    
    3. If a student submits a written test, the intermediate written tests will be considered canceled.
    
    Oral tests. During the oral exam, the commission asks questions about the entire program. In particular, the knowledge of the definitions of the main concepts, and of the statements and proofs of the most important results will be evaluated, and the ability to carry out exercises will be verified.

    Exam schedule

    Date Time Location Type Notes
    23/01/2023 09:00 GENOVA Scritto riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    24/01/2023 09:00 GENOVA Orale riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    13/02/2023 09:00 GENOVA Scritto riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    14/02/2023 09:00 GENOVA Orale riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    26/06/2023 09:00 GENOVA Scritto
    28/06/2023 09:00 GENOVA Orale
    17/07/2023 09:00 GENOVA Scritto
    19/07/2023 09:00 GENOVA Orale
    01/09/2023 09:00 GENOVA Scritto
    04/09/2023 09:00 GENOVA Orale
    23/01/2023 09:00 GENOVA Scritto riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    24/01/2023 09:00 GENOVA Orale riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    13/02/2023 09:00 GENOVA Scritto riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    14/02/2023 09:00 GENOVA Orale riservato agli studenti iscritti a.a.2021/2022 e anni accademici precedenti
    26/06/2023 09:00 GENOVA Scritto
    28/06/2023 09:00 GENOVA Orale
    17/07/2023 09:00 GENOVA Scritto
    19/07/2023 09:00 GENOVA Orale
    01/09/2023 09:00 GENOVA Scritto
    04/09/2023 09:00 GENOVA Orale

    FURTHER INFORMATION

    Teaching style: in presence.