Menù principale
B016761 - ELEMENTS OF MATHEMATICS AND STATISTICS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2023-24
Course year
First year - First Semester
Belonging Department
Experimantal and Clinical Medicine
Course Type
Single education field course
Scientific Area
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
Credits
9
Teaching Hours
72
Teaching Term
18/09/2023 ⇒ 22/12/2023
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
- Last names A-K PACCOSI MICHELE
- Last names A-K PACIFICI EMANUELE
- Last names L-Z FOCARDI MATTEO
Teaching Language - Last names A-K
Italian
Course Content - Last names A-K
Infinitesimal, differential and integral calculus. Elements of probability and statistics.
Suggested readings - Last names A-K (Search our library's catalogue)
(a) M. Abate, Matematica e Statistica, le basi per le scienze della vita, Ed. Mc Graw Hill Education
(b) P. Marcellini, C. Sbordone, Elementi di calcolo, Ed. Liguori
(c) P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica, I volume, Ed. Liguori
(d) J.R. Taylor,
Introduzione all'analisi degli errori, Zanichelli.
(e) Lecture notes by Prof. R. Ricci downloadable from the link http://web.math.unifi.it/users/ricci/TAIS/metodistat.pdf
(f) Worksheets and written tests from past years are provided by the teacher via moodle.
(b) P. Marcellini, C. Sbordone, Elementi di calcolo, Ed. Liguori
(c) P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica, I volume, Ed. Liguori
(d) J.R. Taylor,
Introduzione all'analisi degli errori, Zanichelli.
(e) Lecture notes by Prof. R. Ricci downloadable from the link http://web.math.unifi.it/users/ricci/TAIS/metodistat.pdf
(f) Worksheets and written tests from past years are provided by the teacher via moodle.
Learning Objectives - Last names A-K
Knowledge and understanding:
Acquisition of the tools and knowledge of Mathematics, with particular regard to elementary Mathematical Analysis (functions of one variable, limits, derivatives, qualitative study, integrals, differential equations).
Acquisition of knowledge and basic tools of statistics.
Skills: students who have passed the exam will be able to represent data or functions in graphical form, analyze a function, calculate simple integrals, evaluate the probability of events, determine the predictive power of a diagnostic test.
Acquisition of the tools and knowledge of Mathematics, with particular regard to elementary Mathematical Analysis (functions of one variable, limits, derivatives, qualitative study, integrals, differential equations).
Acquisition of knowledge and basic tools of statistics.
Skills: students who have passed the exam will be able to represent data or functions in graphical form, analyze a function, calculate simple integrals, evaluate the probability of events, determine the predictive power of a diagnostic test.
Prerequisites - Last names A-K
Basic knowledge of arithmetic, algebra, elementary geometry and trigonometry, at a secondary school level.
Teaching Methods - Last names A-K
Traditional lectures and exercise class.
Additional teaching material will be provided via the course moodle page.
Additional teaching material will be provided via the course moodle page.
Further information - Last names A-K
Attendance of lectures and exercises: not compulsory
Teaching support tools: whiteboard, projector, moodle e-learning platform
Reception hours
Prof. Pacifici: by appointment at DiMaI study 012.
Address: DiMaI "Ulisse Dini"
Viale G.B. Morgagni, 67 / A - 50134 Florence
E-mail: emanuele.pacifici@unifi.it
Teaching support tools: whiteboard, projector, moodle e-learning platform
Reception hours
Prof. Pacifici: by appointment at DiMaI study 012.
Address: DiMaI "Ulisse Dini"
Viale G.B. Morgagni, 67 / A - 50134 Florence
E-mail: emanuele.pacifici@unifi.it
Type of Assessment - Last names A-K
The exam consists of a compulsory written test and an optional oral exam.
In the written test, the student will have to solve both infinitesimal, differential and integral calculus exercises, and set up problems of probability and statistics, and give the proof of some theorems among those whose proof was carried out during the course in class (they are indicated on the program).
The exercises are designed to assess students' ability to apply their acquired theoretical and technical knowledge to modeling and problem solving. Both the correctness of the procedures followed and the originality of the methods adopted and their effectiveness are evaluated with particular attention.
Admission to the oral exam is achieved with a grade greater than or equal to 16/30. The oral exam is optional for students who have obtained a mark greater than or equal to 18/30 in the written test (unless otherwise indicated by the teacher), while it is compulsory for those who have obtained the marks 16-17 / 30.
During the oral exam some questions on exercises will be asked to verify the knowledge and the degree of understanding of the theory developed in the course. The ability to communicate the subject in a critical way and the use of an appropriate mathematical language will also be evaluated with particular attention.
In the written test, the student will have to solve both infinitesimal, differential and integral calculus exercises, and set up problems of probability and statistics, and give the proof of some theorems among those whose proof was carried out during the course in class (they are indicated on the program).
The exercises are designed to assess students' ability to apply their acquired theoretical and technical knowledge to modeling and problem solving. Both the correctness of the procedures followed and the originality of the methods adopted and their effectiveness are evaluated with particular attention.
Admission to the oral exam is achieved with a grade greater than or equal to 16/30. The oral exam is optional for students who have obtained a mark greater than or equal to 18/30 in the written test (unless otherwise indicated by the teacher), while it is compulsory for those who have obtained the marks 16-17 / 30.
During the oral exam some questions on exercises will be asked to verify the knowledge and the degree of understanding of the theory developed in the course. The ability to communicate the subject in a critical way and the use of an appropriate mathematical language will also be evaluated with particular attention.
Course program - Last names A-K
1 - NUMBERS AND REAL FUNCTIONS.
Numerical Sets. Basics of set theory. Concept of a real function of a real variable and its Cartesian representation. Invertible functions. Monotone functions. Properties and graphs of elementary functions (linear functions, absolute value, powers, exponentials, logarithms, rational functions, trigonometric functions and their inverses). Solving methods for equations and inequalities. Geometric loci.
2 - LIMITS OF SEQUENCES AND FUNCTIONS, CONTINUOUS FUNCTIONS.
Definition of limit of a sequence. Operations with limits. Indeterminate forms. Comparison theorem. bounded sequences. Notable limits. The number e. Definition of limit of a function. Operations with function limits. Notable limits. Infinitesimal and infinite. Continuous functions. Continuity of elementary functions. Properties. Classification of discontinuities. Absolute maximums and minimums. Continuous function theorems: sign permanence, existence of zeros and intermediate values, Weierstrass theorem, invertibility criterion.
3 - DIFFERENTIAL CALCULUS.
Definition of derivative. Derivative of the sum, of the product, of the quotient. Derivative of the composite function and of the inverse function. Derivatives of
elementary functions. Equation of the tangent line to the graph of a function. Relative maximums and minimums. Fermat's theorem. Rolle's and Lagrange's theorems. Monotony criterion. Convex and concave functions; convexity criterion. Flexes. L'Hopital's theorem. Vertical, horizontal, oblique asymptotes. Study of the graph of a function.
4 - INTEGRAL CALCULUS.
Definition of the integral according to Riemann of a continuous function in an interval. Properties. Mean theorem. Primitives. Characterization of the primitives of a function in an interval. Fundamental theorem of integral calculus. Definition and properties of indefinite integrals. Methods of indefinite integration by decomposition in sum, by parts and by substitution. Integration of rational functions. Calculation of areas of plane figures.
5 - DIFFERENTIAL EQUATIONS.
First order differential equations: linear, Bernoulli, with separable variables. Cauchy's problem. Growth model of an isolated population. Linear differential equations of the second order with constant coefficients.
6 - ELEMENTS OF PROBABILITY AND STATISTICS.
Basics of combinatorics. Events. Introduction to the concept of probability. Axioms of probability. Conditional probability, Bayes' theorem. Independent events. Random variables. Distribution of probability, mean value, variance and standard deviation of a random variable. Discrete random variables: binomial distribution, Poisson distribution. Continuous random variables: uniform distribution, normal distribution
Numerical Sets. Basics of set theory. Concept of a real function of a real variable and its Cartesian representation. Invertible functions. Monotone functions. Properties and graphs of elementary functions (linear functions, absolute value, powers, exponentials, logarithms, rational functions, trigonometric functions and their inverses). Solving methods for equations and inequalities. Geometric loci.
2 - LIMITS OF SEQUENCES AND FUNCTIONS, CONTINUOUS FUNCTIONS.
Definition of limit of a sequence. Operations with limits. Indeterminate forms. Comparison theorem. bounded sequences. Notable limits. The number e. Definition of limit of a function. Operations with function limits. Notable limits. Infinitesimal and infinite. Continuous functions. Continuity of elementary functions. Properties. Classification of discontinuities. Absolute maximums and minimums. Continuous function theorems: sign permanence, existence of zeros and intermediate values, Weierstrass theorem, invertibility criterion.
3 - DIFFERENTIAL CALCULUS.
Definition of derivative. Derivative of the sum, of the product, of the quotient. Derivative of the composite function and of the inverse function. Derivatives of
elementary functions. Equation of the tangent line to the graph of a function. Relative maximums and minimums. Fermat's theorem. Rolle's and Lagrange's theorems. Monotony criterion. Convex and concave functions; convexity criterion. Flexes. L'Hopital's theorem. Vertical, horizontal, oblique asymptotes. Study of the graph of a function.
4 - INTEGRAL CALCULUS.
Definition of the integral according to Riemann of a continuous function in an interval. Properties. Mean theorem. Primitives. Characterization of the primitives of a function in an interval. Fundamental theorem of integral calculus. Definition and properties of indefinite integrals. Methods of indefinite integration by decomposition in sum, by parts and by substitution. Integration of rational functions. Calculation of areas of plane figures.
5 - DIFFERENTIAL EQUATIONS.
First order differential equations: linear, Bernoulli, with separable variables. Cauchy's problem. Growth model of an isolated population. Linear differential equations of the second order with constant coefficients.
6 - ELEMENTS OF PROBABILITY AND STATISTICS.
Basics of combinatorics. Events. Introduction to the concept of probability. Axioms of probability. Conditional probability, Bayes' theorem. Independent events. Random variables. Distribution of probability, mean value, variance and standard deviation of a random variable. Discrete random variables: binomial distribution, Poisson distribution. Continuous random variables: uniform distribution, normal distribution