Differential and integral calculus for functions of one varible. Ordinary differential equations.
Course Content - Last names M-Z
1) Real Numbers; 2) Limits and continuity for one variable functions; 3) Differential Calculus; 4) Integral calculus; 5) Series; 6) Ordinary Differential Equations.
Bertsch - Dal Passo - Giacomelli, Analisi Matematica, second edition, McGraw-Hill.
Learning Objectives - Last names A-L
Practical and theoretical knowledge of basic tools of calculus.
Prerequisites - Last names A-L
Standard knowledge in mathematics, learned in high school. In particular: basic notions on real numbers. Exponential, logarithmic and trigonometric functions.
Teaching Methods - Last names A-L
Theoretical lectures and exercise sessions.
Type of Assessment - Last names A-L
Written and oral exam.
Course program - Last names A-L
1) BASIC NOTIONS. Real numbers; natural, integer and rational numbers. Decimal representations of real numbers. Absolute value. Supremum and infimum of subsets of real numbers. The Dedekind axiom.
Functions: the notion of function. The image and the graph of a function. Injectivity; surjectivity. Inverse function. Composition of functions.
Odd and even functions. Monotone functions. Periodic functions. Elementary functions.
2) LIMITS AND CONTINUITY. Limits of sequences and functions. Operations with limits. Comparison theorems. Limits of monotone functions. The number e. The notion of continuity. Operations with continuity. Continuity of elementary functions. The fundamental theorems for continuous functions: existence of zeroes and intermediate values. Weierstrass' theorem.
3) DIFFERENTIAL CALCULUS. The derivative and its geometrical meaning. Differentiability and continuity. Derivatives of elementary functions. Operations with derivatives. Derivative of the inverse functions and of the composition of two functions. The theorems of Fermat, Rolle and Lagrange. Local and global extrema and their identification. Second derivative and its relation with concavity and convexity. The study of the graph of a function.
Del'Hospital theorems. Taylor polynomial and its applications; Taylor expansions of some elementary functions.
4) INTEGRALS. The definition of integral through Riemann sums and its geometrical meaning. Classes of integrable functions. Basic properties of
the integral. The integral function of a function. The mean value theorem. The fundamental theorem and the fundamental formula of integral calculus.
Indefinite integrals and primitives. Some integration techniques; integration by parts; integration by change of variable.
5) INFINITE SERIES. The notion of series. Partial sums of a series; character of a series. Necessary condition for the convergence. The harmonic and geometric series. Series with non-negative terms. Convergence criteria for series with non-negative terms. Series with arbitrary signs. Absolute convergence; Leibnitz criterion.
6) ORDINARY DIFFERENTIAL EQUATIONS. Ordinary differential equations and the Cauchy problem. First order equations; some technique of solution: linear equations; separation of variables; Bernoulli equations. Linear differential equations of second order, with constant coefficients: general resulution procedure.