Learning outcomes of the course unit
The goal is to provide the students tools and methods in order to treat with limits, differential calculus and integral calculus for real functions of one real variable, ODE and elements of mathematical Statistic.
Also, judgement independence, strong written and oral communication skills, learning ability, in accordance with the specific objects of the mathematical area.
Abilty to handle basic mathematical expressions and to deal with mathematical equalities and inequalities.
Course contents summary
Basic notions of set theory and mathematical logic. Real numbers. Real functions of a real variable and their properties. Limit, continuity, differentiability, and Riemann integral. ODE. Brief introduction to Probability and Statistics.
1.Basic notions of mathematical logic. Basic notions of set theory.
2.Integers, rational numbers, irrational numbers. Upper bound, maximum, least upper bound (supremum). The completeness axiom. Functions and terminology concerning functions. Composite functions. One-to-one functions and inverses. Elementary functions and their diagrams : absolute value, rational, exponential, logarithmic, power with real exponent, trigonometric.
3.Limit of a function. One-side limit of a function. Properties of the limits of functions. Continuous functions. Theorems concerning continuous functions on an interval.
4.Probability Theory: events, classical definition of Probability, combinatorics.
5.Statistics: mean, mode, median. Variance, standard deviation. Regression line; Pearson coefficient. T Test. \Chi^2 Test.
6.Definition of derivative. Derivatives and continuity. Algebra of derivatives. The chain rule. One-sided derivatives and infinite derivatives. Zero derivatives and local extrema. Rolle's theorem. The mean-value theorem for derivatives. Higher order derivatives. Taylor's formula with remainder. Convexity of a function. Diagram of a function.
7.Definition of the Riemann integral. Linear properties. Integration by parts. Change of variable in a Riemann integral. Mean value for the Riemann integral. The integral as a function of the interval. Fundamental theorems of integral calculus. Generalized integrals and comparison theorems.
8.Differential equations. Cauchy problems. Linear first order equations. Separable variables equations. Second oreder linear equations with constant coefficients.
M. Abate, "Matematica e Statistica", McGraw Hill.
M. Bramanti, C. D. Pagani, S. Salsa, "Analisi Matematica 1", Ed. Zanichelli.
A. Guerraggio, "Matematica per le scienze", Ed. Pearson.
P. Marcellini, C.Sbordone, "Elementi di Analisi Matematica Uno", Ed. Liguori.
The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are selected so that the student will be able to solve independently many related problems arising from the theoretical lessons.
Assessment methods and criteria
Final written pretest (25min)+test(2h), and possibly in an oral discussion.