## CALCULUS ANALISI MATEMATICA

A.Y. Credits
2015/2016 12
Lecturer Email Office hours for students
Raffaella Servadei Tuesday 14-16 or by appointment

### Assigned to the Degree Course

Date Time Classroom / Location

### Learning Objectives

Aim of the course is to give to the students some basic tools and topics in mathematical analysis, for functions both with one and several variables.

### Program

01. Numbers:
01.01 Natural numbers, integers, rational numbers and real numbers.
01.02 Summations, factorials, binomial coefficients and Newton's binomial formula.
01.03 Algebraic properties and geometric representation of rational numbers.
01.04 From rational numbers to real ones.
01.05 Absolute value and distance in the real line.
01.06 Intervals in the real line. Bounded and unbounded sets in the real line. Maximum and minimum of a subset of the real line. Supremum and infimum of a subset of the real line.
01.07 Mathematical induction and its applications.

02. Functions of one variable:
02.01 What is a function.
02.02 Real functions of one real variable: preliminaries, bounded functions, even and odd functions, monotone functions, periodic functions.
02.03 Elementary functions.
02.04 Scaling and shifting a graph.
02.05 Piecewise defined functions.
02.06 Composition of functions.
02.07 Inverse function.
02.08 Inverse trigonometric functions.

03. Limits of functions:
03.01 Finite limits at a point.
03.02 Uniqueness theorem for limits*.
03.03 Finite limits at infinity.
03.04 Horizontal asymptotes.
03.05 Infinite limits at infinity.
03.06 Oblique asymptotes. Infinite limits at a point.
03.07 One-sided limits.
03.08 Vertical asymptotes.
03.09 Non-existence of a limit.
03.10 Algebra of limits and indeterminate forms.
03.11 Sign-stability theorem*.
03.12 The squeeze theorem*.
03.13 Change of variable in the limits.
03.14 Asymptotic functions.
03.15 Notable special limits.
03.16 Comparison between infinite functions.

04. Sequences:
04.01 Definition of sequence.
04.02 Convergent, divergent and indefinite sequences.
04.03 Monotone sequences.

05. Continuity:
05.01 Continuous functions.
05.02 Algebra of continuous functions.
05.03 Continuity of elementary functions.
05.04 Continuity of the composition of functions.
05.05 Limits for polynomial functions.
05.06 Limits for rational functions.
05.07 Notable special limits.
05.08 Points of discontinuity.
05.09 Continuous functions on an interval: Bolzano Theorem (bisection method)*, Weierstrass Theorem and Intermediate-Value Theorem*.
05.10 Continuity of the inverse function.

06. Differential calculus for functions of one variable:
06.01 Derivative of a function.
06.02 Geometric interpretation of the derivative.
06.03 Tangent line to the graph of a function.
06.04 Derivatives of elementary functions.
06.05 Relations between differentiability and continuity*.
06.06 Algebra of derivatives*.
06.07 Differentiation rules for product and quotient*.
06.08 The chain rule*.
06.09 Derivative of the inverse function*.
06.10 One-sided derivative and non-differentiability points.
06.11 Critical points, local and global maxima and minima.
06.12 Fermat Theorem*.
06.13 Mean-Value Theorem* and its applications: monotonicity test and characterization of functions with zero derivative in an interval.
06.14 Finding maxima and minima of a function.
06.15 De L'Hospital Rule.
06.16 Second-order derivatives.
06.17 Concavity and convexity of a function.
06.18 Inflection points.
06.19 Sketching the graph of a function.

07. Integral calculus for functions of one variable:
07.01 Antiderivative and indefinite integral of a function.
07.02 Antiderivatives of elementary functions.
07.03 Areas of plane regions.
07.04 Definition of definite integral.
07.05 Integrable functions.
07.06 Properties of definite integral.
07.07 A Mean-Value Theorem for integrals*.
07.08 The fundament theorem of integral calculus*.
07.09 Techniques of integration: scomposition and substitution.
07.10 Integrals of rational functions.
07.11 Integration by parts*.
07.12 Integrals of trigonometric functions.
07.13 Integration of irrational functions.
07.14 Improper integrals.
07.15 Integrability criteria: comparison and limit comparison*.

08. Series:
08.01 Definition and examples: geometric series, harmonic series and generalized harmonic series.
08.02 Necessary condition for the convergence of a series*.
08.03 Convergence tests for positive series: comparison test* and limit comparison test*, root test* and ratio test*.
08.04 Relation between series and improper integrals.
08.05 Absolute convergence of a series.
08.06 Alternating series.
08.07 The alternating series test (Leibnitz test).
08.08 Series depending on a parameter.

09. Approximation of functions and Taylor's formula:
09.01 Differential of a function and linear approximation.
09.02 "small-o" notation.
09.03 Approximating values of functions and application to the calculus of limits.
09.04 Taylor polynomials.
09.05 Taylor's formula with Peano remainder*.
09.06 Taylor's formula for elementary functions.
09.07 Taylor's formula with Lagrange remainder and integral remainder.
09.08 Applications: approximations of functions, error estimates and calculus of limits.
09.09 Taylor series and Taylor series of elementary functions.

10. Differential calculus for functions of several variables:
10.01 Preliminaries.
10.02 Domain, graph and level curves.
10.03 Topology in Rn: distance and its properties, neighborhood, open and closed sets and their properties.
10.04 Limits.
10.05 Calculus of limits: restrictions method and polar coordinates.
10.06 Continuity.
10.07 Weierstrass Theorem.
10.09 Geometric meaning of partial derivatives.
10.10 Directional derivatives.
10.11 Tangent plane.
10.12 Differentiability and linear approximation.
10.13 C^1 functions are differentiable*.
10.14 Relation between directional derivatives and gradient of a differentiable function.
10.15 Algebra of derivatives.
10.16 Chain rule.
10.17 Higher-order derivatives.
10.18 Schwarz Theorem*.
10.19 Taylor's formula in R2.

11. Curves in Rn:
11.01 Vector-valued functions.
11.02 Limits and continuity.
11.03 Derivatives for vector-valued functions.
11.04 Chain rule for vector-valued functions.
11.05 Curves in R2 and in Rn.
11.06 Parametric representation of a curve.
11.07 Closed curves and simple curves.
11.08 Curves in the plane and in the space.
11.09 Parametrization of curves in the plane: lines and line segments.
11.10 Parametric equations of conics.
11.11 Graphs of functions.
11.12 Parametrizations of curves in the space.
11.13 Smooth curves.
11.14 Tangent vector.
11.15 Piecewise smooth curves.

12. Optimization:
12.01 Maxima and minima for functions of one or several variables.
12.02 Critical points.
12.03 First order necessary condition (Fermat Theorem).
12.04 Hessian matrix.
12.05 Classification of critical points in R2 and in Rn.
12.06 Constrained maxima and minima of a function.
12.07 The method of Lagrange multipliers.

13. Integral calculus for functions of two variables:
13.01 Double integrals over a rectangle.
13.02 Iteration formula over a rectangle.
13.03 Geometric meaning of double integrals.
13.04 Double integrals over more general domains.
13.05 Properties of the double integral.
13.06 Iteration of double integrals in Cartesian coordinates.
13.07 Change of variables in R2.
13.08 Change of variables in double integrals.
13.09 Improper double integrals.

14. Ordinary differential equations:
14.01 Preliminaries.
14.02 Separable equations.
14.03 Linear differential equations: preliminaries and superposition principle.
14.04 First-order linear equations.
14.05 Homogeneous constant-coefficients linear equations of higher-order: solution.
14.06 Non-homogeneous constant-coefficients linear equations of higher-order: solution.
14.07 Euler equation and Bernoulli equation.
14.08 Non-linear differential equations.
14.09 Cauchy problem.
14.10 Local existence and uniqueness for the Cauchy problem.
14.11 Boundary value problems.

* : this means that the proof is required.

### Bridging Courses

Although there are no mandatory prerequisites for this exam, students are strongly recommended to take it after Discrete Structures and Linear Algebra.

It is also worth noticing that the topics covered by this course will be used in Algorithms and Data Structures, Digital Signal and Image Processing, Physics I and Probability and Statistics.

It is recommended to take the exam of Calculus during the first year of the Laurea Degree Program in Applied Computer Science.

### Learning Achievements (Dublin Descriptors)

Knowledge and understanding:
At the end of the course the student will learn the basic notions of mathematical analysis for the study of functions of one and several variables.

Applying knowledge and understanding:
At the end of the course the student will learn the methodologies of mathematical analysis and will be able to apply them to the study of various problems.

Making judgements:
At the end of the course the student will be able to apply the techniques of mathematical analysis in order to solve new problems, also coming from real-world applications.

Communications skills:
At the end of the course the student will have the ability to express the fundamental notions of mathematical analysis using a rigorous terminology.

Learning skills:
During the course the student will learn the ability to study the notions of mathematical analysis, also in order to use it in solving different kind of problems.

### Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

### Teaching, Attendance, Course Books and Assessment

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Adams, Calcolo Differenziale 1, Casa Editrice Ambrosiana
Adams, Calcolo Differenziale 2, Casa Editrice Ambrosiana
Barutello - Conti - Ferrario - Terracini - Verzini, Analisi matematica, Vol.2, Apogeo
Bramanti - Pagani - Salsa, Analisi matematica 1, Zanichelli
Bramanti - Pagani - Salsa, Analisi matematica 2, Zanichelli
Bramanti - Pagani - Salsa, Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli
Conti - Ferrario - Terracini - Verzini, Analisi matematica, Vol.1, Apogeo
Salsa - Squellati, Esercizi di Analisi matematica 1, Zanichelli
Salsa - Squellati, Esercizi di Analisi matematica 2, Zanichelli

Assessment

The exam of Calculus consists of a written exam and an oral one, both of them mandatory.
The written exam, to carry out in three hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.
The oral exam consists of a discussion related to the topics of the course. The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.
The final mark of Calculus is the average of the marks of the written exam and the oral one.

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.

### Notes

The course is offered also on-line inside the Moodle platform > elearning.uniurb.it

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