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B019481 - MATHEMATICAL ANALYSIS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Sustainable Development Goals 2030
Academic Year 2022-23
Coorte 2022 - 3-years First Cycle Degree (DM 270/04) in CIVIL, BUILDING AND ENVIRONMENTAL ENGINEERING
Course year
First year - First Semester
Belonging Department
Civil and Environmental Engineering (DICEA)
Course Type
Single education field course
Scientific Area
MAT/05 - MATHEMATICAL ANALYSIS
Credits
9
Teaching Hours
81
Teaching Term
12/09/2022 ⇒ 16/12/2022
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Teaching Language
ITALIAN
Course Content
1) Real Numbers; 2) Limits and continuity for one variable functions; 3) Differential Calculus; 4) Integral calculus; 5) Series; 6) Ordinary Differential Equations.
Suggested readings (Search our library's catalogue)
C. Pagani, S. Salsa, Analisi Matematica 1, ed. Zanichelli, 2015
S. Salsa, A. Squellati: Esercizi di Analisi Matematica 1, ed. Zanichelli, 2011
S. Salsa, A. Squellati: Esercizi di Analisi Matematica 1, ed. Zanichelli, 2011
Learning Objectives
Provide the skill to connect and express in appropriate mathematical language the topics taught during the lectures. In particular: differential and integral calculus for functions of one real variable; series; ordinary differential equations.
Prerequisites
Basic notions of mathematics taught in high school. In particular: formal calculus; polynomials; polynomial equations of degree one and two; algebraic, exponential and logarithmic equations. Basic notions in analytic geometry (the Cartesian plane, lines circles, parabolas, etc.). Trigonometric functions and their properties.
Teaching Methods
Lectures and exercise classes.
Further information
Type of Assessment
The exam aims to verify, through exercises, theoretical questions, and possibly an oral part:
1. the ability to use in an appropriate way the tools and the terminology of the subject;
2. the ability to state, prove and apply the main notions and theorems provided within the course.
The exam is made of a written test of exercises (or equivalently, two intermediate tests during the course) and a written theoretical test, possibly supplemented by an oral part.
1. the ability to use in an appropriate way the tools and the terminology of the subject;
2. the ability to state, prove and apply the main notions and theorems provided within the course.
The exam is made of a written test of exercises (or equivalently, two intermediate tests during the course) and a written theoretical test, possibly supplemented by an oral part.
Course program
1. Natural, integer, rational and real numbers. Decimal representation of real numbers. Absolute value. The distance between real numbers. Intervals. Inf and sup of sets of real numbers. Axion of continuity and completeness of real numbers. Quantifiers, logical implications, necessary and sufficient conditions.
Notion of function; domain, graph, sup and inf, maxima and minima. Injectivity and surjectivity. Invertibility and inverse functions. Composition of functions. Graphs of elementary functions. Odd, even, monotone, and periodic functions.
2. Sequences and their limit. Limits of functions. Uniqueness. Comparison theorems; limits of monotone functions. Operations with limits. Landau symbols. The number e. Some trigonometric and exponential limits. Continuity; examples of discontinuous functions. Continuity of classes of elementary functions. Continuity and elementary operations. Continuity and composition. Intermediate value theorem; Weierstrass theorem.
3. Incremental ratio and geometric and physical meaning. Definition of derivative; equation of the tangent line. Right and left derivative. Corners, cusps, points with vertical tangent. Derivatives of higher order. Derivative of the elementary functions. Operations with derivatives. Derivative of the composition of functions, and of the inverse function. Fermat's theorem. Critical points. Local and global extrema. Rolle's and Lagrange's theorem. Convexity and concavity, and their relation with first and second derivatives. De l"Hospital rule. Taylor polynomials, with applications.
4. The notion of area of a planar figure. Definition of Riemann integral and Riemann integrability. Classes of integrable functions. Integral function. Definition of primitive function. Fundamental theorem and fundamental formula of calculus. Integration techniques. Integrals over half-lines. Comparison theorems for integrals over half-lines of positive functions.
6. Number series. Partial sums and character of a series. Geometric and harmonic series. Necessary condition for the convergence. Series with non-negative terms. Criteria of convergence. General series; criteria of convergence.
7. Ordinary differential equations. Notion of solution. The Cauchy problem. Analysis of various types of equation of the first order. Linear equation of the second order with constant coefficients. Description of the space of solutions, and resolution techniques.
Notion of function; domain, graph, sup and inf, maxima and minima. Injectivity and surjectivity. Invertibility and inverse functions. Composition of functions. Graphs of elementary functions. Odd, even, monotone, and periodic functions.
2. Sequences and their limit. Limits of functions. Uniqueness. Comparison theorems; limits of monotone functions. Operations with limits. Landau symbols. The number e. Some trigonometric and exponential limits. Continuity; examples of discontinuous functions. Continuity of classes of elementary functions. Continuity and elementary operations. Continuity and composition. Intermediate value theorem; Weierstrass theorem.
3. Incremental ratio and geometric and physical meaning. Definition of derivative; equation of the tangent line. Right and left derivative. Corners, cusps, points with vertical tangent. Derivatives of higher order. Derivative of the elementary functions. Operations with derivatives. Derivative of the composition of functions, and of the inverse function. Fermat's theorem. Critical points. Local and global extrema. Rolle's and Lagrange's theorem. Convexity and concavity, and their relation with first and second derivatives. De l"Hospital rule. Taylor polynomials, with applications.
4. The notion of area of a planar figure. Definition of Riemann integral and Riemann integrability. Classes of integrable functions. Integral function. Definition of primitive function. Fundamental theorem and fundamental formula of calculus. Integration techniques. Integrals over half-lines. Comparison theorems for integrals over half-lines of positive functions.
6. Number series. Partial sums and character of a series. Geometric and harmonic series. Necessary condition for the convergence. Series with non-negative terms. Criteria of convergence. General series; criteria of convergence.
7. Ordinary differential equations. Notion of solution. The Cauchy problem. Analysis of various types of equation of the first order. Linear equation of the second order with constant coefficients. Description of the space of solutions, and resolution techniques.