Calculus and Computer Laboratory

SYLLABUS
· Natural, integer, rational, real numbers. The field of real numbers and its operations. The symbols + ∞ and -∞.
· Real functions of real variable. Properties: injectivity, surjectivity, biunivocity, monotony. Inverse functions. Composition of functions. Cartesian representation of the graph of a function.
· Elementary functions: powers, logarithms, exponentials, trigonometric functions, absolute value.
· Linear algebra: vectors, matrices and their operations. Determinant of a square matrix. Inverse matrix. Rank of a matrix. Systems of linear equations and matrix representation. Cramer theorem and Rouchè-Capelli theorem.
· Limits of functions: definitions and first properties. Uniqueness of the limit. Limits of monotone functions. Limits of elementary functions. Operations with limits. Indeterminate forms. Asymptotic functions. Comparison theorems.
· The number of Nepero e. Special limits. Hierarchy of infinities and infinitesimals. Continuous functions and their properties: zeros theorem and Weierstrass theorem.
· Differential calculus: definition of derivative; geometrical meaning; tangent line. Derivability and continuity. Operations with derivatives. Derivatives of composition of functions and inverse functions. Derivatives of elementary functions. Applications of derivatives. Extreme points.
· Fermat's theorem. Rolle theorem and Lagrange theorem. Increasing and decreasing functions. Convexity. De l'Hôpital theorem.
· Study of the graph of a function. Horizontal, vertical, oblique asymptotes.
· Indefinite integral. Calculus of primitives: integration by sum decomposition, integration by parts, integration by substitution. Integration of rational functions (outline).
· Definite integral. Definition, geometric interpretation, properties. The integral mean value theorem. The fundamental theorem of integral calculus. The formula of integral calculus theorem. Area of plane regions.
· Outline of linear differential equations of first and second order.

REFERENCE MATERIAL
· P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore
· D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Casa Editrice Ambrosiana
OR
D. Benedetto, M. Degli Esposti, C. Maffei: Dalle funzioni ai modelli, Casa Editrice Ambrosiana
· Corso online "Matematica Assistita", http://ariel.unimi.it/User/


PREREQUISITES
Elementary algebra, analytic geometry, plane trigonometry. Elementary functions and their graphs. Inequalities.
(see also the MiniMat course material available online at http://ariel.unimi.it/User/).

EXAMINATION PROCEDURES
The exam consists of two parts, one for each of the two modulus of Matematica Generale (6 CFU) and of Laboratorio di Informatica (3 CFU).
To participate to the test of each modulus, the student must register through SIFA to the module itself.
The final grade is obtained after passing the tests related to the two modulus.
The two tests must be passed within the same academic year.
The grade resulting from the weighted average of the grades reported in the two modulus (approximated to the unit by default / excess depending on the decimal number 0-4 / 5-9) will be registered by the professor of the modulus of Matematica Generale.

To pass the exam relating to the modulus of Matematica Generale it is necessary to know the theory (definitions, theorem statements) presented during the lectures and to be able to solve the types of exercises illustrated during the exercises lectures.
The test of the modulus of Matematica Generale consists of exercises and some multiple choice questions on the contents of the theory. The test is considered passed if at least 18 points are obtained out of a total of 30.
There will be 7 tests distributed during the academic year, dates appearing on SIFA.
The written test can be replaced by passing two in itinere tests, the first indicatively in mid-November, the second in the same date of the January test.
The two in itinere tests are passed if in each one a score of at least 16 points is obtained out of a total of 30 and if the average of the two scores (approximated by excess to the unit) is greater or equal to 18.
Candidates must attend the written tests or in itinere tests with a valid identity document with a photograph during which it is not allowed to consult any type of material, nor the use of calculators.
In case the student wants to improve the grade obtained in the written test or in the itinere tests or to obtain the Laude, the student must also pass an oral exam that will focus on the detailed program indicated on the website at the end of the semester (definitions, statements of theorems, proofs of theorems).
In the event of a negative evaluation of the oral exam, the mark obtained in the written test could be modified accordingly or even the written test will have to be repeated.


TEACHING METHODS
Traditional. Tutoring activity for the preparation of the written tests.

FREQUENCY
Strongly recommended.

LANGUAGE
Italian.

PROGRAM INFORMATION
Additional information on the program available on the webpage.

WEBpage
http://ariel.unimi.it/User/

·Natural, integer, rational, real numbers. The field of real numbers and its operations. The symbols + ∞ and -∞.
·Real functions of real variable. Properties: injectivity, surjectivity, biunivocity, monotony. Inverse functions. Composition of functions. Cartesian representation of the graph of a function.
·Elementary functions: powers, logarithms, exponentials, trigonometric functions, absolute value.
·Linear algebra: vectors, matrices and their operations. Determinant of a square matrix. Inverse matrix. Rank of a matrix. Systems of linear equations and matrix representation. Cramer theorem and Rouchè-Capelli theorem.
·Limits of functions: definitions and first properties. Uniqueness of the limit. Limits of monotone functions. Limits of elementary functions. Operations with limits. Indeterminate forms. Asymptotic functions. Comparison theorems.
·The number of Nepero e. Special limits. Hierarchy of infinities and infinitesimals. Continuous functions and their properties: zeros theorem and Weierstrass theorem.
·Differential calculus: definition of derivative; geometrical meaning; tangent line. Derivability and continuity. Operations with derivatives. Derivatives of composition of functions and inverse functions. Derivatives of elementary functions. Applications of derivatives. Extreme points.
·Fermat's theorem. Rolle theorem and Lagrange theorem. Increasing and decreasing functions. Convexity. De l'Hôpital theorem.
·Study of the graph of a function. Horizontal, vertical, oblique asymptotes.
·Indefinite integral. Calculus of primitives: integration by sum decomposition, integration by parts, integration by substitution. Integration of rational functions (outline).
·Definite integral. Definition, geometric interpretation, properties. The integral mean value theorem. The fundamental theorem of integral calculus. The formula of integral calculus theorem. Area of plane regions.
·Outline of linear differential equations of first and second order.

Reference material:
·P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore
·D. Benedetto, M. Degli Esposti, C. Maffei: Matematica per le scienze della vita, Casa Editrice Ambrosiana
OR
D. Benedetto, M. Degli Esposti, C. Maffei: Dalle funzioni ai modelli, Casa Editrice Ambrosiana
·Corso online "Matematica Assistita", http://ariel.unimi.it/User/


Prerequisites:

Elementary algebra, analytic geometry, plane trigonometry. Elementary functions and their graphs. Inequalities.
(see also the MiniMat course material available online at http://ariel.unimi.it/User/).

Examination procedures:

The exam consists of two parts, one for each of the two modulus of Matematica Generale (6 CFU) and of Laboratorio di Informatica (3 CFU).
To participate to the test of each modulus, the student must register through SIFA to the module itself.
The final grade is obtained after passing the tests related to the two modulus.
The two tests must be passed within the same academic year.
The grade resulting from the weighted average of the grades reported in the two modulus (approximated to the unit by default / excess depending on the decimal number 0-4 / 5-9) will be registered by the professor of the modulus of Matematica Generale.

To pass the exam relating to the modulus of Matematica Generale it is necessary to know the theory (definitions, theorem statements) presented during the lectures and to be able to solve the types of exercises illustrated during the exercises lectures.
The test of the modulus of Matematica Generale consists of exercises and some multiple choice questions on the contents of the theory. The test is considered passed if at least 18 points are obtained out of a total of 30.
There will be 7 tests distributed during the academic year, dates appearing on SIFA.
The written test can be replaced by passing two in itinere tests, the first indicatively in mid-November, the second in the same date of the January test.
The two in itinere tests are passed if in each one a score of at least 16 points is obtained out of a total of 30 and if the average of the two scores (approximated by excess to the unit) is greater or equal to 18.
Candidates must attend the written tests or in itinere tests with a valid identity document with a photograph during which it is not allowed to consult any type of material, nor the use of calculators.
In case the student wants to improve the grade obtained in the written test or in the itinere tests or to obtain the Laude, the student must also pass an oral exam that will focus on the detailed program indicated on the website at the end of the semester (definitions, statements of theorems, proofs of theorems).
In the event of a negative evaluation of the oral exam, the mark obtained in the written test could be modified accordingly or even the written test will have to be repeated.

Teaching Methods:
Traditional. Tutoring activity for the preparation of the written tests.
Frequency
Strongly recommended.

Language:
Italian.

Program information:
Additional information on the program available on the webpage.

Website:
http://www.mat.unimi.it/users/cecilia/DIDATTICAGEN.html