60
Mathematical Analysis
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
The course “Elements of Basic Mathematics” aims to introduce students, in a simple and clear way, to mathematical language, providing the necessary tools to understand the core subjects of their study program.
It seeks to help students acquire a solid understanding and command of the main concepts of Mathematical Analysis, Linear Algebra, and Analytical Geometry, which are considered essential for design disciplines and their practical applications.
It also aims to promote the ability to independently and consciously use theoretical knowledge to approach, formulate, and solve even complex problems.
Furthermore, the course encourages critical thinking and independent judgment in the use of mathematical tools, while fostering the development of adequate communication skills and enhancing individual learning abilities.
Course Prerequisites
Knowledge of the main mathematical concepts and techniques covered in upper secondary school programs.
Teaching Methods
- TYPE OF EDUCATIONAL ACTIVITIES:
The course, in order to achieve the expected learning objectives, is mainly carried out through lectures.
In addition, there are exercises conducted by the instructor and guided practice sessions carried out by the students.
Lectures: 40 hours
Exercises: 20 hours
- INDEPENDENT STUDENT WORK
1 CFU = 25 hours (10 hours of lectures / 15 hours of independent study*)
- In-depth study based on the bibliography (theoretical part)
- Exam preparation
Assessment Methods
The exam consists of a final written test and an optional oral exam, which is accessed only if the student achieves at least a minimum score in the final written test.
Passing any ongoing written tests exempts the student from the final written test, or part of it.
Passing the written test entitles the student to take the oral exam only in the session in which the written test was passed.
The written test assesses the student's critical thinking skills in framing the course topics and the methodological rigor of the proposed solutions to the questions.
This test has a maximum duration of two hours. The oral exam consists of an interview covering the topics of the written test and
the theoretical topics included in the course syllabus. The student's ability to communicate the acquired knowledge through appropriate scientific language and presentation skills are assessed.
To pass the exam with the minimum grade of 18/30, students must demonstrate at least an elementary level of knowledge and competence.
A grade between 19/30 and 24/30 is awarded to students who correctly complete the written part but have only elementary competence in the theoretical part.
A grade between 25/30 and 30/30 is awarded to students who perform well in the written test and show good theoretical understanding.
Students who demonstrate excellent competence in both parts may receive the grade “30 with honors.”
Texts
M. Bramanti, C.D. Pagani, S. Salsa, MATEMATICA - Calcolo infinitesimale e Algebra lineare, 2 ed., Zanichelli, Bologna.
P. Marcellini - C.Sbordone - Esercitazioni di Matematica vol. I, parti 1 e 2 - Ed. Liguori, Napoli.
Contents
1_DESCRIPTION
The course Elements of Basic Mathematics provides students of the Design Degree Program with the essential scientific tools needed to approach design processes in a conscious and rigorous way.
The course develops logical and quantitative skills useful for understanding proportions, structures, and geometric relationships, promoting an analytical approach that integrates the creative dimension of design with the necessary technical precision.
2_COURSE CONTENT
SET THEORY:
Concept of a set. Numerical sets. Subsets of a set. Power set. Operations between sets. The numerical sets N, Z, Q, R. Order and representation on the real line. Intervals.
REVIEW OF:
Algebraic, fractional, irrational, logarithmic, exponential, and absolute value equations and inequalities. Powers with integer and rational exponents, and powers with real base and exponent.
Trigonometry: Measurement of arcs and oriented angles. Sine, cosine, and tangent of an oriented arc. Fundamental relations. Addition, double-angle, half-angle, and prosthaphaeresis formulas.
ELEMENTS OF ANALYTIC GEOMETRY:
Lines and oriented segments. Cartesian coordinates in the plane. Distance between two points. Midpoint of a segment. Equation of a line in implicit and explicit form. Slope and its geometric meaning. Conditions for parallelism and perpendicularity between two lines. Distance from a point to a line.
Equations of the circle, parabola, ellipse, and hyperbola. Intersection between a conic and a line. Intersection between curves. Tangent line to a conic.
REAL FUNCTIONS OF A REAL VARIABLE:
Concept of function. Injective, surjective, and bijective functions. Inverse and composite functions. Elementary functions. Bounded, unbounded, monotone, and periodic functions. Infimum and supremum of functions. Absolute maxima and minima.
CONTINUITY OF REAL FUNCTIONS OF A REAL VARIABLE:
Definition of limit. Right-hand and left-hand limits. Existence of the limit. Asymptotes. Algebra of limits. Indeterminate forms. Uniqueness theorem of limits. Sign preservation theorem. Comparison theorem. Notable limits. Limits of monotone functions. Infinitesimals and infinities. Substitution principle. Definition of continuity. Types of discontinuity and their classification. Continuity of composite functions. Weierstrass theorem. Intermediate value theorem. Invertibility criterion. Zero existence theorem. Continuity of the inverse function.
DIFFERENTIAL CALCULUS FOR FUNCTIONS OF ONE VARIABLE:
Definition of derivative and geometric meaning. Corner points and cusps. Derivatives of elementary functions. Operations with derivatives. Differentiability and continuity. Chain rule. Derivative of the inverse function and related applications. Higher-order derivatives. Increasing and decreasing functions. Relative maxima and minima. Critical points. Fermat’s theorem. Rolle’s theorem, Lagrange’s theorem, Cauchy’s theorem. Geometric interpretation and consequences of Lagrange’s theorem. De L’Hôpital’s rule. Concavity and convexity. Inflection points. Graph analysis of a function.