# Computer science | MATHEMATICAL ANALYSIS

## Computer science MATHEMATICAL ANALYSIS

 0512100001 DIPARTIMENTO DI INFORMATICA EQF6 COMPUTER SCIENCE 2016/2017

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2015 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 48 LESSONS BASIC COMPULSORY SUBJECTS MAT/05 3 24 EXERCISES BASIC COMPULSORY SUBJECTS

 SARA MONSURRO' T
Objectives
KNOWLEDGE AND UNDERSTANDING: TO PROVIDE STUDENTS WITH THE BASIC NOTIONS OF CALCULUS: TO BE ABLE TO SOLVE A RANGE OF SIMPLE EXERCISES. IN PARTICULAR, STUDENTS WILL BE ABLE TO DRAW THE GRAPHIC OF A FUNCTION GIVEN ITS ALGEBRAIC FORM AND TO CALCULATE SIMPLE INTEGRALS.
APPLYING KNOWLEDGE AND UNDERSTANDING: LEARN CALCULUS THEORY AND APPLICATIONS.
TO BE ABLE TO PRESENT, IN A CLEAR AND RIGOROUS MANNER, THE THEOREMS, DEFINITIONS AND PROBLEMS STUDIED THROUGHOUT THE COURSE.
Prerequisites
ELEMENTARY ASPECTS OF ALGEBRA ARE MANDATORY AS WELL AS FAMILIARITY WITH SOLUTION METHODS FOR FIRST AND SECOND ORDER EQUALITIES AND INEQUALITIES. KNOWLEDGE OF SOME ELEMENTS OF TRIGONOMETRY ARE ALSO CONSIDERED A PREREQUISITE.
Contents
REAL NUMBERS

THE AXIOMS OF REAL NUMBERS AND THEIR CONSEQUENCES. NATURAL NUMBERS, INTEGERS AND RATIONAL NUMBERS. NUMERICAL SETS: BOUNDED SETS, MAXIMUM AND MINIMUM, SUPREMUM AND INFIMUM. UNIQUENESS OF THE MINIMUM (WITH PROOF). EXISTENCE THEOREM FOR THE SUPREMUM OF A NUMERICAL SET (WITH PROOF). UNBOUNDED SETS. INTERVALS. ARCHIMEDEAN PROPERTY (WITH PROOF). NON-COMPLETENESS OF THE RATIONAL NUMBERS (WITH PROOF).

REAL FUNCTIONS OF A REAL VARIABLE

DEFINITION AND NOTATIONS. EXTREMA. CARTESIAN REPRESENTATION. ODD FUNCTIONS, EVEN FUNCTIONS AND PERIODIC FUNCTIONS. COMPOSITION OF FUNCTIONS. INVERSE FUNCTIONS. MONOTONE FUNCTIONS AND THEIR INVERSE (WITH PROOF) .

ELEMENTARY FUNCTIONS

LINEAR FUNCTIONS. THE ABSOLUTE VALUE. TRIANGUAR INEQUALITY. POWER FUNCTIONS. LOGARITHM AND EXPONENTIAL FUNCTIONS. TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS.
NUMERICAL SEQUENCES

DEFINITION AND NOTATIONS. CONVERGING SEQUENCES: DEFINITION AND EXAMPLES. UNIQUENESS OF THE LIMIT (WITH PROOF). DIVERGENT SEQUENCES: DEFINITIONS AND EXAMPLES. IRREGULAR SEQUENCES. BOUNDEDNESS OF CONVERGENT SEQUENCES (WITH PROOF). OPERATIONS WITH CONVERGING SEQUENCES (PROOFS OF THE SUM AND THE PRODUCT). THEOREM OF THE PERMANENCE OF THE SIGN AND COROLLARIES (WITH PROOFS). THEOREM OF THE “CARABINIERI” (WITH PROOF). THEOREM OF THE PRODUCT OF A BOUNDED SEQUENCE TIMES AN INFINITESIMAL ONE. MONOTONE SEQUENCES AND THEIR THEOREM (WITH PROOF). THE NUMBER E. NOTABLE SPECIAL LIMITS. EXTRACTED SEQUENCES. THEOREM OF BOLZANO - WEIERSTRASS .
LIMITS OF FUNCTIONS

DEFINITION AND NOTATIONS. LINK BETWEEN LIMITS OF FUNCTIONS AND LIMITS OF SEQUENCES . UNIQUENESS OF THE LIMIT. OPERATIONS WITH LIMITS. INDETERMINATE FORMS. LIMITS OF COMPOSITE FUNCTIONS. LIMITS OF THE ELEMENTARY FUNCTIONS AT THE EXTREMA OF THEIR DOMAIN. NOTABLE SPECIAL LIMITS (WITH PROOFS).

CONTINUOUS FUNCTIONS

DEFINITION. CONTINUITY OF ELEMENTARY FUNCTIONS. OPERATIONS WITH CONTINUOUS FUNCTIONS. COMPOSITION OF CONTINUOUS FUNCTIONS. SIGN PERMANENCE THEOREM (WITH PROOF). MEAN VALUE THEOREM. WEIERSTRASS EXISTENCE THEOREM.

DERIVATIVES

DEFINITION AND GEOMETRICAL MEANING. CONTINUITY OF DIFFERENTIABLE FUNCTIONS (WITH PROOF). OPERATIONS WITH DERIVATIVES (WITH PROOFS). DERIVATIVE OF A COMPOSITION OF FUNCTION. DERIVATIVE OF THE INVERSE FUNCTION. DERIVATIVES OF THE ELEMENTARY FUNCTIONS (WITH PROOFS). HIGHER ORDER DERIVATIVES. THEOREMS OF DE L'HOPITAL. TAYLOR'S FORMULA .

APPLICATIONS OF DERIVATIVES. STUDY OF REAL FUNCTIONS OF A REAL VARIABLE

MAXIMA AND MINIMA OF A REAL FUNCTION OF ONE REAL VARIABLE. FERMAT'S THEOREM (WITH PROOF). ROLLE'S THEOREM (WITH PROOF). LAGRANGE'S THEOREM (GRAPHICAL INTERPRETATION). MONOTONICITY CRITERION (WITH PROOF). STRICT MONOTONICITY CRITERION. CHARACTERIZATION OF CONSTANT FUNCTIONS IN AN INTERVAL (WITH PROOF). CONVEX AND CONCAVE FUNCTIONS. CRITERION OF CONVEXITY. SUFFICIENT CONDITIONS FOR THE DETERMINATION OF THE RELATIVE EXTREMA OF A FUNCTION. STUDY OF A FUNCTION’S GRAPH.
RIEMANN INTEGRATION

DEFINITION IN THE CASE OF A BOUNDED FUNCTION AND NOTATIONS. GEOMETRIC INTERPRETATION. ADDITIVITY AND LINEARITY PROPERTIES. INTEGRABILITY OF CONTINUOUS FUNCTIONS. THEOREMS OF THE INTEGRAL MEAN VALUES (WITH PROOFS) .

INDEFINITE INTEGRALS

FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS (WITH PROOF). CHARACTERIZATION OF THE PRIMITIVE OF A FUNCTION IN AN INTERVAL (WITH PROOF). FUNDAMENTAL FORMULA OF THE INTEGRAL CALCULUS (WITH PROOF). INDEFINITE INTEGRATION AND ITS PROPERTIES. INTEGRATION BY SUM DECOMPOSITION. INTEGRATION OF RATIONAL FUNCTIONS. INTEGRATION BY PARTS. SOME TYPES OF INTEGRATION BY SUBSTITUTION.
Teaching Methods
• FRONTAL LESSON
• PRACTICE LESSON
Verification of learning
THE KNOWLEDGE AND UNDERSTANDING OF THE TOPICS DESCRIBED WITHIN THE COURSE WILL BE TESTED BY MEANS OF A FINAL WRITTEN EXAMINATION CONTAINING THEORY AND EXERCISES TO TEST THE KNOWLEDGE OF THE BASIC NOTIONS OF CALCULUS (TO BE ABLE TO DRAW THE GRAPHIC OF A FUNCTION GIVEN ITS ALGEBRAIC FORM AND TO CALCULATE SIMPLE INTEGRALS). THE WRITTEN EXAMINATION WILL BE FOLLOWED BY AN ORAL EXAM TO VERIFY THAT THE STUDENT IS ABLE TO PRESENT, IN A CLEAR AND RIGOROUS MANNER, THE THEOREMS, DEFINITIONS AND PROBLEMS STUDIED THROUGHOUT THE COURSE.
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