ID:
20121
Dettaglio:
SSD: Mathematical Analysis
Duration: 72
CFU: 9
Located in:
DALMINE
Url:
BUILDING TECHNOLOGY ENGINEERING - 20-R-TE/Geometra Laureato Year: 1
ENGINEERING FOR SUSTAINABLE ENERGY AND ENVIRONMENTAL TECHNOLOGY - 174-R/PERCORSO COMUNE Year: 1
Year:
2025
Overview
Date/time interval
Primo Semestre (15/09/2025 - 20/12/2025)
Syllabus
Course Objectives
At the end of the course the student will master the main methods and techniques of mathematical analysis. In particular, she/he will be able to:
1. compute limits and derivatives, use these tools to study the behavior of real functions of one real variable and then draw a qualitative graph;
2. use the main techniques for the determination of the primitive of a function and then compute definite integrals;
3. use the basic notions of complex numbers and linear algebra;
4. apply linear algebra to the study of geometry in three dimensions.
In order to know the potential and limitations of the tools described above, the student will also have a full awareness of their theoretical foundations and will express them with an adequate language.
1. compute limits and derivatives, use these tools to study the behavior of real functions of one real variable and then draw a qualitative graph;
2. use the main techniques for the determination of the primitive of a function and then compute definite integrals;
3. use the basic notions of complex numbers and linear algebra;
4. apply linear algebra to the study of geometry in three dimensions.
In order to know the potential and limitations of the tools described above, the student will also have a full awareness of their theoretical foundations and will express them with an adequate language.
Course Prerequisites
1. Plane Euclidean geometry: in particular, triangle criteria for equality and similarity, Euclid and Pythagoras theorems, elementary properties of polygons and circles. One-to-one correspondence between real numbers and points on a line; intervals, half line; Cartesian plane; distance between two points in the plane. Elementary loci in the plane: line (parallelism and orthogonality conditions), circle. ellipse, parabola and hyperbola.
2. Equations and Inequalities of first and second degree. System of equations and inequalities.
3. Powers with integer exponent, properties of powers. Polynomials: divisibility, Ruffini rule, roots, factorization. Powers with rational and real exponent, graphs and main properties. Exponential and logarithmic function: graph and their main properties.
4. Irrationals equations and inequalities; equations and inequalities with exponentials, logarithms and absolute value.
5. Trigonometry: measure of angles in radiants; fundamental identity; graphs of sine, cosine and tangent. Equations and inequalities with trigonometric functions.
6. Real function of one real variable: domain, codomain, graph; intersection between graphs. Absolute value function; graphs of f(-x), di f(|x|), di |f(x)|, di f(x+c), di f(x)+c. Even, odd and periodic functions.
2. Equations and Inequalities of first and second degree. System of equations and inequalities.
3. Powers with integer exponent, properties of powers. Polynomials: divisibility, Ruffini rule, roots, factorization. Powers with rational and real exponent, graphs and main properties. Exponential and logarithmic function: graph and their main properties.
4. Irrationals equations and inequalities; equations and inequalities with exponentials, logarithms and absolute value.
5. Trigonometry: measure of angles in radiants; fundamental identity; graphs of sine, cosine and tangent. Equations and inequalities with trigonometric functions.
6. Real function of one real variable: domain, codomain, graph; intersection between graphs. Absolute value function; graphs of f(-x), di f(|x|), di |f(x)|, di f(x+c), di f(x)+c. Even, odd and periodic functions.
Teaching Methods
The teaching is composed by lectures and exercises (for a total amount of 72 hours), and tutoring (18 hours). In all three activities the student is encouraged to participate with suggestions and proposals. If remote or blended teaching is necessary, some changes will be introduced, also for the exams.
Assessment Methods
The exam aims to verify the achievement by students of the educational objectives described above. In particular:
- mastery of the methods and techniques developed;
- awareness of their theoretical foundations;
- appropriateness of the language used.
The exam can be taken only by students who have fulfilled their OFA in mathematics. It consists of two parts, a practical one and a theoretical one. The mark takes into account the correctness, clarity and the ability to justify the conclusions. In alternative to the examination procedure described above, as far as the practical part is concerned, students can take the exam with two half-period tests. Even students that have not satisfied their OFA in mathematics can access the first test. Instead, the access to the second test (which takes place the same day of the full exam in January or in February) requires to have fulfilled the OFA.
- mastery of the methods and techniques developed;
- awareness of their theoretical foundations;
- appropriateness of the language used.
The exam can be taken only by students who have fulfilled their OFA in mathematics. It consists of two parts, a practical one and a theoretical one. The mark takes into account the correctness, clarity and the ability to justify the conclusions. In alternative to the examination procedure described above, as far as the practical part is concerned, students can take the exam with two half-period tests. Even students that have not satisfied their OFA in mathematics can access the first test. Instead, the access to the second test (which takes place the same day of the full exam in January or in February) requires to have fulfilled the OFA.
Contents
1 NUMBERS
Sets. Summations, geometrical progression. Real numbers. Maximum and minimum. Supremum and infimum. Powers and roots. Exponential and logarithm. Complex numbers. Functions.
2 ELEMENTS OF GEOMETRY AND LINEAR ALGEBRA
Vectors in the plane and in the space. Space analytical geometry. The space R^n. Matrices.
3 SEQUENCES AND SERIES
Convergent, divergent, and irregular sequences; monotone sequences; computation of limits. Convergent, divergent, and irregular series.
4 FUNCTIONS OF A SINGLE VARIABLE: LIMITS AND CONTINUITY
Graph of a function; bounded, odd, even, monotone, and periodic functions. Limits, continuity, and asymptotes of a function. Function composition; invertible functions.
5 DIFFERENTIAL CALCULUS FOR A FUNCTION OF A SINGLE VARIABLE
Derivative of a function. Rules for computing derivatives. Stationary points, local maxima and minima, mean value (or Lagrange's) theorem, monotony test, the search for maxima and minima, de l’Hospital theorem. Second derivative, concavity and convexity. Drawing of the graph of a function. Differential calculus and approximations: little-o notation, Taylor-MacLaurin formula with the Peano and the Lagrange form of the remainder.
6 INTEGRAL CALCULUS FOR FUNCTIONS OF A SINGLE VARIABLE
The integral as a limit of sums. The fundamental theorem of integral calculus. Computation of indefinite and definite integrals: decomposition, substitution, and by parts methods. Integral functions; second fundamental theorem of integral calculus.
Sets. Summations, geometrical progression. Real numbers. Maximum and minimum. Supremum and infimum. Powers and roots. Exponential and logarithm. Complex numbers. Functions.
2 ELEMENTS OF GEOMETRY AND LINEAR ALGEBRA
Vectors in the plane and in the space. Space analytical geometry. The space R^n. Matrices.
3 SEQUENCES AND SERIES
Convergent, divergent, and irregular sequences; monotone sequences; computation of limits. Convergent, divergent, and irregular series.
4 FUNCTIONS OF A SINGLE VARIABLE: LIMITS AND CONTINUITY
Graph of a function; bounded, odd, even, monotone, and periodic functions. Limits, continuity, and asymptotes of a function. Function composition; invertible functions.
5 DIFFERENTIAL CALCULUS FOR A FUNCTION OF A SINGLE VARIABLE
Derivative of a function. Rules for computing derivatives. Stationary points, local maxima and minima, mean value (or Lagrange's) theorem, monotony test, the search for maxima and minima, de l’Hospital theorem. Second derivative, concavity and convexity. Drawing of the graph of a function. Differential calculus and approximations: little-o notation, Taylor-MacLaurin formula with the Peano and the Lagrange form of the remainder.
6 INTEGRAL CALCULUS FOR FUNCTIONS OF A SINGLE VARIABLE
The integral as a limit of sums. The fundamental theorem of integral calculus. Computation of indefinite and definite integrals: decomposition, substitution, and by parts methods. Integral functions; second fundamental theorem of integral calculus.
Online Resources
Degrees
Degrees (2)
BUILDING TECHNOLOGY ENGINEERING - 20-R-TE
Bachelor's Degree
3 years
ENGINEERING FOR SUSTAINABLE ENERGY AND ENVIRONMENTAL TECHNOLOGY - 174-R
Bachelor's Degree
3 years
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Main module
Mathematical Analysis and Geometry