SM/0025 - MATHEMATICAL ANALYSIS 1
Academic Year 2021/2022
Free text for the University
ANTONIO GRECO (Tit.)
- Teaching style
- Lingua Insegnamento
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1. Knowledge and understanding
Knowledge and understanding of the terms and the notation in the theory of limits and differential and integral calculus for functions of one real variable. Knowledge of the basic notions of the theory of numerical sequences and infinite series, as well as the theory of functions of one real variable. Elements of the theory of ordinary differential equations.
2. Applying knowledge and understanding
Ability in the usage of the algorithms for computing limits of numerical sequences as well as of functions of one real variable, and for the symbolical computation of derivatives and the integration by quadratures of such functions. Usage of integration theory to compute the area of simple plain regions. Usage of some criteria to establish the convergence of an infinite series. Ability to integrate some simple differential equation.
3. Making judgements
Apply the differential calculus to discover the qualitative properties of the graph of a given function, and to determine its maxima and minima. Bring together and connect the information arising from different algorithms concerning a given function, and intelligently use such information to establish the nature of the function. Critically evaluate if an alleged proof, or, more generally, a given argument is correct, as well as if a given definition is well posed. Locate and correct one's own mistakes, without getting lost.
4. Communication skills
Ability in communicating information, ideas, problems and solutions by means of the current terminology, in order to effectively interact with other students and teachers, in the present, and prospectively with colleagues or in a possible teaching activity. Ability in supporting a claim by means of a logical argument (a proof), making a correct usage of terms as "if", "then", "that is to say".
5. Learning skills
Being able to consult and interpret textbooks to build and expand one's own knowledge. Being able to use the sources of information as a tool in problem-solving activities. Being able, if necessary, to use the sources of information in order to go beyond a strict interpretation of the course boundaries.
1. Arithmetic with polynomial and rational expressions (indispensable).
2. Fundamentals of Euclidean geometry in the plane: theorem of Pythagoras, area of a triangle, square, rectangle (important); area of a circle (useful).
3. Basic notions of analytic geometry in the plane: coordinates of a point; equation of the line and meaning of the coefficients (important).
4. Fundamentals of trigonometry: circular functions and their graphs; special values; main relations (useful).
5. Exponential and logarithmic function and their main properties (useful).
1. Infinite sequences of numbers and their limits
Infinite sequences: converging, diverging, irregular, monotonic, bounded; cluster points, Bolzano-Weierstrass theorem; absolute value; triangle inequality; limit of a sequence; operation rules; sign-preserving property; comparison theorem; indeterminate forms; some special sequences. Relation between the least upper bound (supremum) and tha completeness of the set of the real numbers; Euler's number
2. Finite sums and infinite series
The summation notation; sum of the first n positive integers; definition of a convergent, absolutely convergent, divergent, irregular series; definition of the sum of an infinite series; geometric series, harmonic series; convergence tests
3. Elements of the theory of functions of one real variable
Bounded functions; even, odd, monotonic, periodic functions; power functions, exponential and logarithmic functions, trigonometric functions and their graphs; trigonometric identities; operations on graphs; piecewise-defined functions
4. Limits of functions of one variable
Sequential definition of the limit of a function; uniqueness; right-hand and left-hand limit; asymptotes; continuity; non-existing limits; intervals; neighborhoods; interior, exterior, boundary points; open, closed, sequentially compact sets; comparison theorem; sign-preserving property; operations with limits; continuity of elementary functions; limit of a composite function; continuity of composites; limits of polynomials; limits of rational functions; important limits; zeros of a continuous function (Bolzano's theorem, and the intermediate-value theorem); the extreme-value theorem for continuous functions
5. Differential calculus for functions of one variable
Definition of the derivative; geometrical meaning of the derivative; equation of the tangent line; convex and concave functions; convexity and the second derivative; convexity and tangent lines; n-th derivative; differentiation of elementary functions; corner points; cusps; differentiability implies continuity; differentiation rules; Lagrange's mean value theorem; monotonicity test; functions with vanishing derivative; de l'Hopital rule; Taylor's formula; how to detect maxima and minima with or without the differential calculus; points of inflection; application to drawing the graph of a given function
6. Integral calculus for functions of one variable
Definition of the Riemann integral; geometric meaning of the integral; sufficient conditions for integrability; properties of the integral; primitive of a function; relation between two primitives over the same interval; fundamental theorem of the integral calculus; indefinite integral; immediate integration; integration rules; exploiting symmetries when computing integrals; integration of some classes of functions; improper integrals; comparison test
7. Introduction to ordinary differential equations
Classification of the ordinary differential equations. Separation of variables for first-order equations. Linear equations of the first and second order. Special methods for determining a particular solution of a linear, nonhomogeneous equation
The course is composed of 96 lectures. Traditional lectures are intertwined with Socratic lectures (the lecturer asks questions intended to lead students to reach the conclusion on their own). Homeworks are given. Some theoretical concepts are demonstrated by computer graphics or interactive applets. Volunteering students may contribute to the lecture (in presence as well as online) by presenting their point of view.
Students particularly interested in some topic may request the lecturer to go deeper into the subject. In case of need, students may also require the review of some prerequisite in order to attend more proficiently the subsequent lectures.
In addition, anonymous means of communication are provided in the form of anonymous ballots as well as by an interactive presentation software, with the aim to provide information and data suitable to fine-tune the teaching method in response to the students' needs.
The lecturer can always be reached by sending an e-mail to firstname.lastname@example.org
Verification of learning
On the occasion of the final evaluation, students are required to pass a written test (if permitted by the health-protection regulation) composed of ten questions of an elementary naure, to be answered in written form. Only two errors are allowed. Those who pass the test are immediately admitted to the oral exam. Candidates are asked some questions chosen by the committee in the realm of the course contents. Questions are usually of the same kind and the same level of difficulty as those proposed in class. The student must answer in a mixed oral/written form, making use of a blackboard or write pad in order to:
1. Support the complex arguments that are typical of the mathematical thinking by means of the conventional symbols as well as with graphical representations (when possible).
2. Show the ability to manipulate formal expressions by a correct application of the governing rules.
The final score ranges from 18/30 to 30/30 cum laude. Active participation to lectures may increase the score of 1-2 points.
The final score takes the following elements into account:
1. The underlying logic in the solution of the proposed problem;
2. The correctness of the adopted procedure;
3. The correspondence of the proposed solution with the expected abilities;
4. The usage of a convenient language, and in particular the correct usage of terms such as "if", "that is to say" and "then";
5. The ability to take part into a discussion, and the ability to locate and correct possible mistakes without getting lost.
Initiative, critical and personal thinking are appreciated, as well as the creative usage of the student's knowledge.
Ability in the computation of limits and in the differential calculus, knowledge of the graphs of the most elementary functions, and in some minor respect the ability in the computation of simple integrals and in the determination of the character of simple numerical series are necessary conditions to pass the exam, thus getting a final score not less than 18/30. A more thorough achievement of the educational goals listed above corresponds to a proportionally higher score. Particularly brilliant students may be required to solve more involved problems, corresponding to the highest cognitive level of Bloom's taxonomy, in order to get the maximum final score of 30/30 cum laude.
Keep in mind that students differs very much one from the other because each student is characterized by a peculiar combination of the elements listed above. By contrast, the final score consists of a single numerical value and therefore cannot maintain any memory of each concurring element. Consequently, very different students may well achieve an identical score: to make an example, think for instance about a very careful student (element no. 1) which is shy and uncertain when relating to other people (element no. 5), to be compared to an impulsive student, very brilliant and clear in supporting her claims.
M. Bertsch, A. Dall'Aglio, L. Giacomelli.
Epsilon 1 - Primo corso di Analisi Matematica
P. Marcellini, C. Sbordone.
Esercitazioni di matematica. Volume 1, parte prima e parte seconda.
T. M. Apostol. Calculus. Volume 1. Wiley.
R. Courant. Differential and integral calculus. Volume 1. Interscience/Wiley.
R. Courant, H. Robbins. What is mathematics? Oxford University Press.
E. Giusti. Analisi matematica. Volume 1. Boringhieri.
M. Kline. Storia del pensiero matematico. Einaudi.
C. D. Pagani, S. Salsa. Analisi matematica. Volume 1. Zanichelli.
G. B. Thomas, Jr., R. L. Finney. Calculus. Addison-Wesley.
W. F. Trench. Introduction to Real Analysis. Freely available online.
W. Rudin. Principles of mathematical analysis. McGraw-Hill.
For information on specific learning disabilities (SLD) see http://corsi.unica.it/matematica/info-dsa/
Additional material is available at http://people.unica.it/antoniogreco/
The present report conforms to the SUA card. An effort was made to comply with the recommendations of the DISCENTIA project and the guidelines of PQA (the Committee for Quality Control). Educational goals and criteria for final evaluation are specified, respectively, on the basis of the Dublin descriptors and Bloom's taxonomy.
Regolamento didattico: http://corsi.unica.it/matematica/regolamenti/laurea-triennale/
SUA card: http://corsi.unica.it/matematica/assicurazione-della-qualita/documenti-sua-e-riesame-cds/
DISCENTIA project: http://sites.unica.it/qualita/discentia/
Bloom's taxonomy: https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/
Dublin descriptors, in short: http://www.quadrodeititoli.it/descrittori.aspx?descr=172&IDL=2
More on the Dublin descriptors is found in the original paper "A Framework for Qualifications of the European Higher Education Area" by the Bologna Working Group (2005) available online at ECA (European Consortium for Accreditation): http://ecahe.eu/ On the same site see also "The Origin of the Dublin Descriptors" by Dr. M. Leegwater