Teachings
IN/0186  MATHEMATICAL ANALYSIS 1
Academic Year 2016/2017
Free text for the University
 Professor

MONICA MARRAS (Tit.)
 Period

First Semester
 Teaching style

Convenzionale
 Lingua Insegnamento

Informazioni aggiuntive
Course  Curriculum  CFU  Length(h) 

[70/77] CHEMICAL ENGINEERING  [77/00  Ord. 2016] PERCORSO COMUNE  9  90 
[70/78] MECHANICAL ENGINEERING  [78/00  Ord. 2016] PERCORSO COMUNE  9  90 
Objectives
1. Knowledge and understanding. At the end of the course the student will have knowledge of topics concerning real sequences, the infinitesimal calculus of real functions of a real variable and ordinary differential equations.
2. Applying knowledge and understanding. The student will be introduced to the main applications of the theoretical notions of the course, concerning both the solution of mathematical problems and the study of physical problems.
3. Autonomy of judgment. The student will learn how to classify single problems of differential and integral calculus, as well as differential equations, in the right class and to apply the most convenient method.
4. Communicative Skills. Students will acquire the ability to communicate what they learn and elaborate and also to express and argue the choice of methodology with respect to another to solve a mathematical problem.
5. Learning skills. Thanks to the notions acquired in this course, the student will be able to perfect his knowledge of higher mathematics and its applications to engineering.
Prerequisites
Good knowledge of algebra, trigonometry and elementary analytic geometry
Contents
Course contents
Introduction to set theory. Sets of Natural, Integer, Rational Numbers. Real Numbers: definitions, algebraic operations, distance. Subsets of real numbers. Least upper bound, greatest lower bound; maximum and minimum, accumulation, isolated, internal , external, boundary points; bounded, open, closed sets.
Real functions. Domain of definition, graph of elementary functions. Bounded, periodic, symmetric, monotonic, composite and inverse functions. Maxima and minima.
Limit theory. Basic limit theorems.
Continuous functions. The definition of continuity and basic theorems. Types of discontinuities. Weierstrass Theorem. The intermediate value theorem.
Differential calculus. Definition of derivative. The algebra of derivatives. The derivative of polynomials, of rational , exponential , logarithm, trigonometric functions. Geometric interpretation. Higher derivatives. The chain rule. Estreme values of a function. Increasing and decreasing functions, The mean value, Rolle Cauchy, DeL’Hopital theorems. Second derivative test for extrema. Convex and concave functions. Graph. Taylor and MacLaurin formula.
Integration. Antiderivative. Definite integral.Partitions of intervals. Definition of integral by upper and lower integrals. The area of a set. Theory and techniques of integrations. Fundamental theorem of integral calculus. Improper integrals.
Differential equations. Physical motivations. Theminology and notations. First order differential equation: separable equations, linear, Bernoulli and Clairaut equations. Cauchy theorem for the existence and uniqueness of the solution. Linear equations of order n. Wronskian determinant, Liouville theorem. Linear equations with constant coefficients: Lagrange method and special methods for determining a particular solution of the nonhomogeneous equation.
Sequences and infinite series of real number
Limits of sequence. Theorems.
Convergent harmonic geometric and telescopic series. Tests of convergence for series of nonnegative terms. Test for alternating series. Absolute convergence.
Sequences and infinite series of functions
Pointwise Convergence of sequences. Uniform convergence: theorems on continuity, derivability and integrability. Telescopic series. Tests for convergence.
Power series in the set of real numbers. Interval of convergence and sufficient conditions. Series of Taylor e Mac Laurin.
Teaching Methods
Frontal lectures (theory): 56 hours
Frontal lectures (exercises): 34 hours
Verification of learning
A test, consisting on solving several exercises and proving some theorems. An eventual discussion.
Texts
Marco Bramanti, Carlo D. Pagani, Sandro Salsa: Analisi matematica 1. Zanichelli,
Sandro Salsa, Annamaria Squellati: Esercizi di Analisi matematica 1, Zanichelli.