00674 - Mathematics

Learning outcomes

By the conclusion of this course the student should have acquired a basic knowledge of measure theory, algebraic calculus, infinitesimal analysis, probability calculus and statistics. Such knowledge will be useful in various disciplines.

Course contents

UNITS OF MEASURE SYSTEMS:

Units of measure. Multiples and submultiples. Scientific notation, order of magnitude, significant figures, percentages. Errors, propagation of error (i.e. propagation of uncertainty).

POWERS, ROOTS, LOGARITHMS:

Definitions, properties and operations.

EXPONENTIAL AND LOGARITHMIC EQUATIONS

FUNCTIONS:

Real functions of one real variable: domain, codomain, graph. Elementary functions: linear, polynomial, exponential, logarithmic, trigonometric, with absolute values. Characteristics and graphs of elementary functions. Composite and inverse functions.

LIMITS:

Concept and definition of limit: limits of real functions of one real variable. Calculus of limits. Indeterminate forms. Continuity of a function in a point/interval. Points of discontinuity.

DIFFERENTIAL CALCULUS:

Derivative: its definition, geometric meaning (tangent line) and kinematical interpretation (velocity). Fundamental derivatives. Operations. Derivative of composite and inverse functions. Continuous derivatives. Continuity and derivability. Increasing/decreasing/monotone functions in a point/interval. Points of relative/absolute maximum and minimum. Concavity, convexity, flexes. Asymptotes. Study of a function. Differential of a function.

INTEGRAL CALCULUS:

Indefinite integral: definition. Immediate integrations. Methods of integration: by reduction formulae, by substitution, by parts, integration of fractional rational functions. Definite integral: geometric meaning, properties and theorems. Improper integral. Calculus of areas and volumes.

COMBINATORIAL CALCULUS:

Disposizioni, permutazione and combinazioni semplici and con ripetizione. Binomial coefficient and its properties.

PROBABILITY CALCULUS:

Stochastic events, sample space, complementary event, union/intersection event. Definition of probability: classical, frequentistic, subjectivistic (i.e. epistemic). Assiomatic definition of probability. Compatible, incompatible, dependent, independent events. Conditional probability. Bayes' theorem.

ELEMENTARY STATISTICS:

Location parameters: arithmetical/geometrical/weighted mean; mode, median, mathematical expectation (i.e. expected values), quartiles. Dispersion parameters: range, interquartile range, standard deviation, variance, mean square deviation. Aleatory variables: normal, binomial, Poisson's distributions.

Readings/Bibliography

Vinicio Villani, Matematica per discipline bio-mediche, McGraw-Hill, Milano, 1997, 4a edizione.

Notes from the lessons.

Teaching methods

Class lessons with frequent exercises on the most important subjects. Two self-evaluation tests.

Assessment methods

Final written examination. The written test is based on multiple-choice questions.

Teaching tools

Blackboard/ohp, pc, slides, notes prepared by the teacher.

Office hours

See the website of Maria Rita Bertazzoni