37643 - Mathematical Analysis T

Academic Year 2017/2018

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Civil Engineering (cod. 8888)

    Also valid for First cycle degree programme (L) in Environmental Engineering (cod. 9198)

Learning outcomes

At the end of the course, after the final examination, the student should possess basic knowledge relating basic elements of Mathematical Analysis, and in particular relating the real functions of a real variable: interpretation of graphs, limits, derivatives, integration and their meaning. The student will be able to solve the typical problems of Mathematical Analysis concerning the above topics (limits; the Induction Principle; continuity; monotonicity and convexity; Taylor expansions; Riemann definite and indefinite integration; complex numbers; real series).

Course contents

  • Introduction: Properties of the real numbers and of N, Z, Q, R. Real-valued functions of one real variable; injectivity, surjectivity, invertibility, inverse function, composition of function, monotone functions. Elementary functions: basic facts.
  • Limits: Accumulation point, definitions of limit; one/two-sided limits. Elementary properties of limits: unicity, locality. Algebraic properties of the limit, comparison theorems, limits of monotone functions. Indeterminate forms. Landau symbols.
  • Continuity: Definition of a continuous function of one real variable. The Weierstrass Theorem, the Bolzano Theorem and the Intermediate Value Theorem. Continuity of the composition of two continuous functions.
  • Differential calculus and applications: Definition of a differentiable function and of the derivative of a function. The algebra of derivatives. The mean value theorems and their application in the study of the monotonicity of a function. Higher order derivatives. Hopital's Rule. Taylor's formula. Local maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.
  • Integration: Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts.
  • Improper integrals: basic definitions and the comparison theorem.
  • Complex numbers: Definiton of the field of the complex numbers. Algebraic form. Modulus and argument of a complex number. Exponential form of a complex number. De Moivre's formula. Complex roots of a complex number. Algebraic equations in C.
  • Real series: Basic definitions; necessary condition for the convergence of a series; series of nonnegative real numbers: comparison theorems; the root and ratio criteria. Leibniz theorem.
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    Readings/Bibliography

    G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, Volume 1, Zanichelli Editore (Bologna), 2009

    AmsCampus pdf files, regularly uploaded.

    S. Salsa, A. Squellati: Esercizi di Analisi Matematica 1, Zanichelli Editore (Bologna), 2011

    Teaching methods

    Class lessons and exercises. Theoretical lectures; examples and exercises solved during the regular class lectures; exercises left to the student. Exercises regularly published in the institutional AMS campus site.

    Assessment methods

    Written and oral examinations. A detailed program for the oral part will be published in the institutional AMScampus site. The written part of the examination will check the knowledge of ALL the topics presented in the exercises, regularly published on the AMScampus site; in the written part, some questions may be posed on theoretical topics as well. During the oral examination, the student will be asked at least three theorems/proofs/examples/definitions, presented during the lectures.

    Dates:

    3 exams in January/February

    1 in June: 1 in July, 1 in September

    Teaching tools

    Regularly, exercises will be published on-line in the AMS-Campus site. As for the preparation for the oral examination, at the end of the semester, a very-detailed list of the questions for the oral part will be published in the AMS-Campus site.

    Office hours

    See the website of Andrea Bonfiglioli